Derivations of group rings for finite and FC groups
aa r X i v : . [ m a t h . R A ] J a n Derivations of group rings for finite and FCgroups
A.A.Arutyunov ∗† ; L.M.Kosolapov ‡§ February 2, 2021
Abstract
In this paper we establish decomposition theorems for derivations ofgroup rings. We provide a topological technique for studying derivationsof a group ring A [ G ] in case G has finite conjugacy classes. As a result,we describe all derivations of algebra A [ G ] for the case when G is a finitegroup, or G is an FC-group. In addition, we describe an algorithm toexplicitly calculate all derivations of a group ring A [ G ] in case G is finite.As examples, derivations of Z [ S ] and F m D n are considered. Keywords
Derivations, group algebras, group rings, finite groups, FC-groups,
AMS codes
In this paper we solve the derivation problem for group algebras of finite groups.Our results provide constructive description of derivation algebra
Der ( F [ G ]) forthe case when F is a finite field and G is a finite group. We discuss exampleswhen outer derivations algebra is nontrivial, and provide some criterions for itstriviality.Our theorems admit a natural generalization to the case when F is an ar-bitrary commutative unital ring, and G is an FC-group. In these cases ourtechnique provides decomposition theorems for derivation algebra.Our method opens the way for practical applications of derivation algebras.As an example, the explicit description of derivation algebras is useful for con-struction of binary codes (see [19]). ∗ Department of Higher Mathematics, Moscow Institute of Physics and Technology, [email protected] † V.A. Trapeznikov Institute of Control Sciences of RAS, 65 Profsoyuznaya str., Moscow117997, Russia ‡ Higher School of Economics , Moscow, 101000, Russia, [email protected] § Skolkovo Institute of Science and Technology, Moscow, 121205, Russia .1 History of the topic and motivation for this research In this paper we apply the the character technique approach, which is proposedin [3], to the case when group has finite conjugacy classes. As a consequence, thecharacter technique gives tools to prove a decomposition theorem for
Der ( A [ G ]) in case G is an FC-group and A is a commutative unital ring. As a corollary,we prove a theorem describing all inner and outer derivations of a group ringof a finite group. In addition, theorems about character complexes for finiteconjugacy classes may be useful for studying Derivation problem for groupswhich have nontrivial finite conjugacy classes.Derivations of group rings have been a topic for studies since late 1970s.Smith (see [39]) was one of the first to study the derivations in group rings. Incertain papers (see [14, 22, 38]) were studied the properties of group rings whichhave no outer derivations.There is a lot of papers concerning derivations of group rings: derivations ofgroup rings and polynomial rings were studied at [10, 14, 15, 30, 31], derivationsof particular group algebras (incindence, Grassman, Novikov, nilpotent algebras,etc.) were studied in [8, 9, 28], and there is a lot of topics connected withderivations of group rings (see [33, 38]). In recent papers [1, 2, 29], properties ofgeneralized inner derivations, central derivations and derivations of group ringsover finite rings were studied.The derivation problem asks, in essense, whether all derivations in a groupring are inner. This question for Banach algebra L ( G ) was formulated in [20](Q5.6.B, p.746). Results for some special cases can be found in [27, 34].In recent papers [3, 4, 5, 6, 7, 36] a categorical method of studying deriva-tions was introduced. The method is based upon a correspondence betweenderivations and characters (additive functions on groupoid of adjoint action ofa group). This method proved to be useful in studying derivation problem (i.e.finding whether outer derivation space is trivial).Application of characters technique for these problems is described in [3,4, 5]. Moreover, in recent paper [7] character technique proved to be usefulfor studying ( σ, τ ) -derivations. Derivations, and especially ( σ, τ ) -derivations ofgroup rings have various applications in coding theory [12, 19].In general, information on group rings, derivations, endomorphisms of grouprings, Lie rings (Lie algebras over rings) and Lie algebras can be found in [11,23, 25, 26, 37].In our paper we prove the decomposition theorems which provide the wayto explicitly describe all derivations of a finite group ring. This can be usefulfor coding theory applications (see [12, 19]). The decomposition theorems weprove also give rise to the criterion for solving the problem of outer derivations’existence for finite group rings. In addition, our technique is applicable to FC-group rings, which can be useful for studying their derivations.2 .2 Main results Let G be a noncommutative group, A - commutative unital ring. We study the A -module of derivations over a group ring A [ G ] . Note that group ring A [ G ] comprises of finite sum of elements from G with coefficients from A . Thus,endomorphisms of A [ G ] are linear operators, which map finite sums to finitesums. Definition 1.1.
Operator d : A [ G ] → A [ G ] is called an endomophism of groupring A [ G ] , if: • d ( x + y ) = d ( x ) + d ( y ) for ∀ x, y ∈ A [ G ] , • r ∈ A, d ( rx ) = rd ( x ) for ∀ x ∈ A [ G ] . Of course, by the definition of A [ G ] , any such operator d maps finite sums(of group’s elements) to finite sums. Definition 1.2.
Derivations of A [ G ] are defined as endomorphisms of A [ G ] thatare subject to the Leibnitz rule. I.e. d ∈ Der ( A [ G ]) , if d is an endomorphismof A [ G ] and d ( xy ) = d ( x ) y + xd ( y ) for ∀ x, y ∈ A [ G ] Definition 1.3.
Let a ∈ A [ G ] . Adjoint derivation ad a acts on any x ∈ A [ G ] in the following way: ad a ( x ) = [ a, x ] ≡ ax − xa Inner derivations are defined as usual:
Inn = { d = X a ∈ G C a ad a - finite sum } It is a well-known fact (see [26]) that ( Der , [ ., . ]) form a Lie algebra withrespect to commutator, and Inn is an ideal in
Der . This makes quotient Liealgebra correctly defined:
Out := Der / Inn . In the article we consider the case when G is a group with finite conjugacyclasses – so called FC-group (see [37]). In particular G is a finite group. Alsowe will consider that A is an unital commutative ring. Theorem 3.2.
Let G be a finitely generated FC-group, A be a unital com-mutative ring. Then Der ( A [ G ]) ∼ = Inn ( A [ G ]) ⊕ M [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) . Here (see definition 2.12)
Hom Ab is the set of additive homomorphisms fromthe centralizer Z ( u ) of fixed element u ∈ G to the ring A .This result can be specified for description of outer derivation whether G isa finite group. Theorem 4.1.
There is a way to describe all derivations of A [ G ] in case G is finite: . Der ( A [ G ]) ∼ = Inn ( A [ G ]) ⊕ (cid:0) L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) (cid:1) , Out ( A [ G ]) ∼ = L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) . Generally speaking algebra of outer derivations isn’t trivial. But for specialcases spaces
Hom Ab ( Z ( u ) , A ) can be degenerated.First case is easily deduced from well-known results. Corollary 4.2.4. If A is a torsion-free ring, G is a finite group, then allderivations are inner Out ( A [ G ]) = 0 . As a consequence, if A = Z , Q , R , C , then Out ( A [ G ]) = 0 . But the second case apparently was not previously studied and importantfor applications.If the ring A is finite then as an abelian group the following decompositionholds A ∼ = Z p i ⊕ ... ⊕ Z p inn . Also for finite group G we get G/ [ G, G ] ∼ = Z q j ⊕ ... ⊕ Z q jmm be the primary decomposition of G/ [ G, G ] . Theorem 4.3.
For finite group G and finite ring A all derivations Der ( A [ G ]) are inner ( Der ( A [ G ]) = Inn ( A [ G ]) ) if and only if { p , ..., p n }∩{ q , ..., q m } = ∅ . The results obtained can be used in applications, some of which are describedin the section 5. In particular we will consider the case of an algebra Z [ S ] (see results of section 5.2) and F m D n (see section 5.3). The last example isimportant for applications especially for cryptography. So we will give a newcombinatorial way to get such results from [19, 17, 18]. In this section we definte a category of adjoint action similarly to papers [3, 5].A groupoid Γ based on a noncommutative group G is defined in the followingway:• Ob (Γ) = G • Hom ( a, b ) = { ( u, v ) ∈ G × G | v − u = a, uv − = b } for every a , b ∈ Obj (Γ) • Composition of maps ϕ = ( u , v ) ∈ Hom ( a, b ) , ψ = ( u , v ) ∈ Hom ( b, c ) is a map ϕ ◦ ψ ∈ Hom ( a, c ) such that: ϕ ◦ ψ = ( u v , v v ) Hereinafter for x ∈ G , the corresponding conjugacy class is [ x ] = { gxg − | g ∈ G } . 4 efinition 2.1. Define Γ [ u ] as a subgroupoid of Γ , such that: • Ob (Γ [ u ] ) = [ u ] • Hom (Γ [ u ] ) = { ϕ ∈ Hom (Γ) | t ( ϕ ) ∈ [ u ] } .In other words, Γ [ u ] is a full subcategory in Γ . Denote G G = { [ u ] , . . . , [ u n ] } as a set of conjugacy classes in G . Statement.
It is easy to check by direct calculation that: • Two objects are connected in Γ iff they are conjugated elements of G ; • For any morphism ϕ ∈ Hom (Γ) , we have t ( ϕ ) ∈ [ u ] ⇐⇒ s ( ϕ ) ∈ [ u ] ; • For any conjugacy class [ u ] in G G , the corresponding subgroupoid Γ [ u ] isa connected component in Γ ; • Hence Γ can be represented as a disjoint union Γ = G [ u ] ∈ G G Γ [ u ] . Proof of these facts can be found in previous papers [5, 7].The next definition is important (see [3], section 3).
Definition 2.2.
A function χ : Hom (Γ) → A , such that χ ( ϕ ◦ ψ ) = χ ( ϕ ) + χ ( ψ ) , is called a character on Γ . We denote the space (actually, an A -module) of suchcharacters on Γ as X (Γ) . Generally, this is not the same as the space of locally-finite characters.
Let d be a derivation of A [ G ] .For any basis element g in the group algebra A [ G ] , we have d ( g ) = X h ∈ G d hg h (1)where d gh ∈ A – coefficients that depend only on the derivation d . An arbi-trary element u ∈ A [ G ] can be represented as a finite sum u = P g ∈ G λ g g , withcoefficients λ g ∈ A . Then the element d ( u ) can be represented as d ( u ) = X g ∈ G λ g d ( g ) = X g ∈ G X h ∈ G λ g d hg h. (2)5ollowing [3, 5] the correspondence between derivations and characters isgiven by the usual definition: χ d (( u, v )) = d uv . (3)If d ∈ Der , then ∀ g, d ( g ) = P d hg h = P χ (( h, g )) h is a finite sum. So we cansay, that a character χ corresponds to a derivation, if and only if for each g exists just a finite number of elements h , such that χ (( h, g )) = 0 . In otherwords, ∀ g, χ (( h, g )) = 0 for cofinitely many h ’s. Such a character is called alocally finite character. Definition 2.3.
Following papers [3, 5], a 1-character χ such that for every v ∈ G χ ( x, v ) = 0 for almost all x ∈ G is called locally finite. In other words, a character χ is locally finite, if for any g ∈ G , there is onlya finite number of elements x ∈ G , such that χ ( x, v ) = 0 . Definition 2.4.
Define X fin as a set of locally finite 1-characters on the Γ . As an A -module, X fin is obviously isomorphic to Der . However, we needto establish a Lie algebra isomorphism between X fin and Der . Theorem 2.1. If d , d ∈ Der , then their commutator [ d , d ] corresponds to1-character χ [ d ,d ] , which can be calculated by this formula χ [ d ,d ] ( a, g ) = X h ∈ G χ d ( a, h ) χ d ( h, g ) − χ d ( a, h ) χ d ( h, g ) . (4)Denote { χ d , χ d } := χ [ d ,d ] . Proof.
Let g ∈ G , then we have d ( g ) = X h ∈ G χ d ( h, g ) h, d ( g ) = X h ∈ G χ d ( h, g ) h. We are going to calculate the commutator [ d , d ] = d d − d d . It’s easy tosee that d d ( g ) = X h ∈ G χ d ( h, g )( X a ∈ G χ d ( a, h ) a ) ,d d ( g ) = X h ∈ G χ d ( h, g )( X a ∈ G χ d ( a, h ) a ) . Changing the sum order in these expressions we get [ d , d ]( h ) = P a ∈ G ( P h ∈ G χ d ( h, g ) χ d ( a, h ) − χ d ( h, g ) χ d ( a, h )) a. { χ d , χ d } ( a, h ) is a coefficient of a , i.e. { χ d , χ d } ( a, g ) = P h ∈ G χ d ( a, h ) χ d ( h, g ) − χ d ( a, h ) χ d ( h, g ) .This theorem is an analogue of Proposal 4.3 from [6]. Corollary 2.1.1.
The formula (3) defines the canonical isomorphism ( Der , [ · , · ]) ∼ = ( X fin , {· , ·} ) . Recall that
Inn = { d = X a ∈ G C a ad a - finite sum } . The next statement follows from [3], Theorem 2.
Statement.
An adjoint derivation ad a : x [ a, x ] corresponds to the character χ ad a (( h, g )) = , g − h = hg − = a , hg − = a, g − h = a − , g − h = a, hg − = a , else (5)Now we can write down definitions of inner derivation submodule in termsof characters. Definition 2.5.Inn = { d ∈ Der | χ d = X a C a χ ad a - finite sum } We can clearly see that character of any d ∈ Inn is trivial on loops.
Statement.
Here are some basic facts about how trivial-on-loops characters on Γ behave: • If a character on a groupoid is trivial on loops, then it is constant on any
Hom ( a, b ) ; • If a character χ on a groupoid is trivial on loops, then it is a potentialcharacter, i.e. ∃ p : Ob (Γ) → A = C : χ = ∂p . The latest term means that ∀ ϕ ∈ Hom (Γ) , χ ( ϕ ) = p ( t ( ϕ )) − p ( s ( ϕ )) ; • For any identity morphism ( a, e ) its character χ ( a, e ) = 0 ; • If α ∈ Hom ( a, a ) , β ∈ Hom ( b, b ) , ∃ ϕ ∈ Hom ( a, b ) : α = ϕ − ◦ β ◦ ϕ , then ∀ χ ∈ X , χ ( α ) = χ ( β ) . hese facts are all quite obvious and their proofs are neglected for the sake ofconciseness. We have already said above that a trivial-on-loops character may not haveits corresponding derivation be inner. As an example, in some cases (infinite G , A = C ) there are locally finite trivial-on-loops characters, which do notcorrespond to an inner derivation. Counterexamples can be found in [4] (resultfor nilpotent groups in section 3.2 and more specific result for Heisenberg groupin section 3.3). Definition 2.6.
Let us define a submodule of derivations, such that their char-acters are trivial on loops:
Inn ∗ = { d ∈ Der | χ d is trivial on loops } We call such derivations weakly inner . This definition was proposed in [7] (see section 2.5) where this type of deriva-tions was called quasi-inner.It’s easy to see that all inner derivation are weakly inner
Inn ⊆ Inn ∗ . Converse is not true (see [4], results of section 3.3).
Definition 2.7.
We introduce definitions of character modules consisting onlyof trivial-on-loops characters: X = { χ ∈ X | χ is trivial on loops } ; X fin = { χ ∈ X | χ is locally finite and trivial on loops } . Immediately from definition we get that ( X fin , {· , ·} ) is a Lie algebra. Andmoreover ( X fin , {· , ·} ) ∼ = ( Inn ∗ , [ · , · ]) .The following theorem is an analogue of the Theorem 2.2. for semigroupalgebras from [7]. Lemma 2.1.
These conditions on character are equivalent:1. χ is trivial on loops;2. χ is potential;3. χ can be expressed as a (possibly infinite) sum of adjoint derivations’ char-acters: χ = P g ∈ G C g χ ad g , C g ∈ A .Proof. (1 = ⇒ If χ is trivial on loops, then for any [ u ] ∈ G G , fix u ∈ [ u ] , and chooseany value of p ( u ) , for example, p ( u ) = 0 . Then we can define potential p on Γ [ u ] through the following procedure: if ϕ : u → u, u ∈ [ u ] , then p ( u ) = p ( u ) + χ ( ϕ ) . On one hand, if such ϕ exists, then p ( u ) does not depend on the8hoice of ϕ due to properties of trivial on loops characters. On the other hand, ∃ g : u = gu g − . Hence ∃ ( gu , g ) : u → u , which is an example of such ϕ above.As a result, we obtain p defined on the whole Γ , and χ = ∂ ( p ) . (2 = ⇒ We can clearly see now that ∀ ϕ ∈ Hom (Γ) χ d ( ϕ ) = p ( t ( ϕ )) χ ad t ( ϕ ) ( ϕ ) + χ ad s ( ϕ ) p ( s ( ϕ )) = p ( t ( ϕ )) − p ( s ( ϕ )) . This gives us the correspondence between C g and the values of p . (3 = ⇒ ∀ g, χ ad g is trivial on loops. Any linear combination of trivial-on-loops char-acters is trivial on loops. Lemma 2.2. Inn ∗ is an ideal in Der .Proof.
At first, let d ∈ Der , d = ad x It is a well-known fact that [ d, ad g ]( x ) = ad d ( g ) ( x )[ d, ad g ]( x ) = d ( ad g ( x )) − ad g ( d ( x )) = d ( gx − xg ) − [ g, d ( x )] == d ( g ) x + gd ( x ) − d ( x ) g − xd ( g ) − [ g, d ( x )] = [ d ( g ) , x ] + [ x, d ( g )] = ad d ( g ) ( x ) . Let d ∈ Inn ∗ , i.e. χ d is trivial on loops. Then χ d = P g ∈ G C g χ ad g , hence d can be expressed in the form of an arbitrary sum d = P g ∈ G C g ad g - anarbitrary sum of adjoint derivations (which, nevertheless, maps any element of G to a finite sum). Then [ d, d ]( x ) = [ d, X g ∈ G C g ad g ]( x ) = X g ∈ G C g [ d, ad g ]( x ) = X g ∈ G ad d ( g ) ( x ) . We know that for ∀ g , ad d ( g ) has a trivial-on-loops character. Hence [ d, d ] istrivial on loops. Definition 2.8.
Let χ ∈ X (Γ) . We define support of character χ as supp ( χ ) = { ϕ ∈ Hom (Γ) | χ ( ϕ ) = 0 } Now we can look at character submodules consisting only of characters sup-ported on some particular subgroupoid.
Definition 2.9. • Der [ u ] = { d ∈ Der | supp ( χ d ) ⊂ Γ [ u ] } ; • X (Γ [ u ] ) = { χ ∈ X (Γ) | supp ( χ ) ⊂ Γ [ u ] } ; • X fin (Γ [ u ] ) = { χ ∈ X (Γ) | supp ( χ ) ⊂ Γ [ u ] , χ is locally finite } ; • X fin (Γ [ u ] ) = { χ ∈ X (Γ) | supp ( χ ) ⊂ Γ [ u ] , χ is locally finite and trivial on loops } . A = C . Neverthe-less, all properties of characters on subgroupoids coincide in both cases, withtheir proofs totally identical to the case when A = C .) Statement.
Some trivial facts: • X (Γ [ u ] ) is a submodule in X (Γ) , X fin (Γ [ u ] ) is a submodule in X fin (Γ) .Similarly for X and X fin ; • Der [ u ] ∼ = X fin (Γ [ u ] ) as A -modules; • As a Lie algebra w.r.t. commutator,
Der [ u ] is not a subalgebra in Der .In addition, if X ∈ Der [ u ] , Y ∈ Der [ v ] , then [ X, Y ] is not always ; • Let G be finite . It follows that there is only finite number of objects andmorphisms in Γ . Since a function over a disjoint union of a finite numberof subdomains can be represented in a form of a direct sum, we have thefollowing A -module isomorphisms:1. X (Γ) = L [ u ] ∈ G G X (Γ [ u ] ) ,2. X (Γ) = L [ u ] ∈ G G X (Γ [ u ] ) ,3. It follows that Der ( A [ G ]) ∼ = L [ u ] ∈ G G Der [ u ] . Other facts about derivations on subgroupoids can be found in [3, 4].
Consider an A -module complex from the character modules −→ X fin (Γ [ u ] ) ∂ −→ X fin (Γ [ u ] ) ∂ −→ X fin ( Hom ( u, u )) −→ . (6)The first differential ∂ : X fin (Γ [ u ] ) → X fin (Γ [ u ] ) is defined as an identicalinclusion of functions on Γ .The second differential ∂ : X fin (Γ [ u ] ) → X fin ( Hom ( u, u )) is a projectionon a group of loops over u∂ ( χ )( ϕ ) = ( χ ( ϕ ) , ϕ ∈ Hom ( u, u )0 , else Talking simply, ∂ is just a restriction of character on a Hom ( u, u ) . Theorem 2.2.
The sequence −→ X fin (Γ [ u ] ) ∂ −→ X fin (Γ [ u ] ) ∂ −→ X fin ( Hom ( u, u )) (7) is an exact sequence of A -modules. roof. Firstly, ∂ ◦ ∂ = 0 . Since a restriction of a trivial-on-loops character ona loop group on some particular vertex is zero.Secondly,we will prove that the sequence (7) is exact:1. ker( ∂ ) = { } , since if character is trivial on loops, and equals 0 on non-loops, then it vanishes everywhere;2. im ∂ ≡ X fin ;3. Let us prove that ker ∂ ⊂ im ∂ . By definition, χ ∈ ker ∂ ⇐⇒ χ islocally-finite, and ∀ ϕ ∈ Hom ( u, u ) , χ ( ϕ ) = 0 . It follows that χ is trivialon any loop in Γ [ u ] due to character properties (2.3). Hence χ ∈ im ∂ .Contrary to exactness of the sequence (7), the sequence (6) is not alwaysexact.However, we can prove that sequence of the form (6) becomes exact in case [ u ] has finite size. There are similar character modules and a similar sequence of characters fornon locally-finite characters. In case A = C this sequence has been studied inpapers [3, 4]. Definition 2.10. • X (Γ) is a set of 1-characters on groupoid Γ ; • X (Γ) is a set of trivial on loops 1-characters on groupoid Γ . Then there is a (generally, non-exact) sequence of characters. −→ X (Γ [ u ] ) ∂ −→ X (Γ [ u ] ) ∂ −→ X ( Hom ( u, u )) −→ . (8)For some particular classes of groups G , this sequence coincides with (7). Inaddition, it can be described in terms of cohomology of 2-category (2-groupoid)defined similarly to Γ . Details on this approach can be found in paper [6] (seeTheorem 2.1 and Proposal 4.4). In this section we study the A -module of characters over Γ [ u ] , when the con-jugacy class [ u ] is finite. This provides some techinques, which are useful forstudying Der ( A [ G ]) in case the group G is a finite group or an FC-group. (De-tailed description of FC-groups and their properties can be found in [37]).In addition, these techniques provide a way to solve a Derivation problem(i.e. find out whether Out ( A [ G ]) = 0 ) in some cases when G has at least onefinite conjugacy class. A special example is the case when G is an FC-group,i.e. all conjugacy classes of G are finite.11 emma 2.3. Let g ∈ G , and let | u | < ∞ . Then there are only finitely manymorphisms of the form ( ∗ , g ) in Hom (Γ [ u ] ) .Proof. For any g ∈ G , | h : ( h, g ) ∈ Hom (Γ [ u ] ) | = |{ h : g − h, hg − ∈ [ u ] }| =( since hg − ∈ [ u ] ⇐⇒ g − hg − g = g − h ∈ [ u ])= |{ h : g − h ∈ [ u ] }| = |{ h : h ∈ g [ u ] }| = | g [ u ] | = | [ u ] | < ∞ . In terms of paper [5] (section , p.7), this means | [ u ] | < ∞ = ⇒ |H g ∩ Γ [ u ] | < ∞ . Corollary 2.2.1. If | u | < ∞ , then any character with support in Γ [ u ] is auto-matically locally finite, i.e. X fin (Γ [ u ] ) ≡ X (Γ [ u ] ) , X fin (Γ [ u ] ) ≡ X (Γ [ u ] ) .Proof. Let a character χ be supported in Γ [ u ] . Then for any g we know that allmorphisms of the form ( ∗ , g ) , such that χ ( ∗ , g ) = 0 , lie in in Hom (Γ [ u ] ) . Dueto the lemma above, there can only be a finite number of such morphisms in Hom (Γ [ u ] ) . Thus, χ is locally finite.The following lemma is an important technical tool in this paper. Its con-sequences will be used later to prove decomposition theorems for derivations offinitely generated FC-group rings and finite group rings. Lemma 2.4.
Let the conjugacy class [ u ] be finite. Then the sequence −→ X fin (Γ [ u ] ) ∂ −→ X fin (Γ [ u ] ) ∂ −→ X fin ( Hom ( u, u )) −→ (9) is a split exact sequence of A -modules.Proof. Exactness part
For exactness, we need only im ∂ = X fin ( Hom ( u, u )) .So, let χ u ∈ X fin ( Hom ( u, u )) . Then we can define χ ∈ ∂ − ( χ u ) in thefollowing way: χ ( ϕ ) = χ u ( ϕ ) if ϕ ∈ Hom ( u, u ) , if ϕ is not a loop in Γ[ u ] ,χ u ( θ − ◦ ϕ ◦ θ ) if ϕ ∈ Hom ( u ′ , u ′ ) . (10)In the last case, θ is any morphism in Hom ( u ′ , u ) . Obviously, due to char-acters’ properties (2.3), χ ( ϕ ) does not depend on the choice of θ . In addition, χ ( ϕψ ) = χ u ( θ − ϕψθ ) = χ u ( θ − ϕθθ − ψθ ) == χ u ( θ − ϕθ ) + χ u ( θ − ψθ ) = χ ( ϕ ) + χ ( ψ ) χ is defined correctly. We have to prove that χ is locally finite (on Γ [ u ] ). Let g ∈ G , then there must be finite number of h ’s, such that ( h, g ) ∈ Hom (Γ [ u ] ) and χ (( h, g )) = 0 .On one hand, ( h, g ) ∈ Hom (Γ [ u ] ) = ⇒ g − h, hg − ∈ [ u ] .On the other hand χ (( h, g )) = 0 = ⇒ ( h, g ) is a loop = ⇒ g − h = hg − .We get that |{ h : h ∈ Z ( g ) , hg − ∈ [ u ] }| = |{ h : h ∈ Z ( g ) ∩ [ u ] g }| ≤ | [ u ] g | < ∞ hence χ is locally-finite.2. Split part
Denote id X fin ( Hom ( u,u )) as the identity map on X fin ( Hom ( u, u )) . Thedefinition above defines a section for ∂ , i.e. the map f : χ u χ , has theproperty: ∂ ◦ f = id X fin ( Hom ( u,u )) . Hence the sequence (7) splits. Definition 2.11.
Denote the map described above (10) as f : χ u χ . Thismap will be used later Corollary 2.2.2.
It follows that
Der [ u ] is isomorphic to X fin (Γ [ u ] ) ⊕ X fin ( Hom ( u, u )) . Corollary 2.2.3. If | u | < ∞ , then −→ X (Γ [ u ] ) ∂ −→ X (Γ [ u ] ) ∂ −→ X ( Hom ( u, u )) −→ (11) is a split exact sequence of A -modules. Here the differentials are same as forthe sequence above. Definition 2.12.
Let H be some group, A be a ring. We denote Hom Ab ( H, A ) as the set of additive homomorphisms from the group H to the ring A . Lemma 2.5.
The following isomorphism holds X ( Hom ( u, u )) ∼ = Hom Ab ( Z ( u ) , A ) . Proof.
It is easy to notice that
Hom ( u, u ) is a group with respect to morphismcomposition. Obviously, this group is canonically isomorphic to centralizer sub-group Z ( u ) . In other words, characters on Hom ( u, u ) are nothing but additivehomomorphisms from Z ( u ) to A . Recall that a group G is called an FC-group (see [37]), if all conjugacy classesof G are of a finite size. 13 emma 3.1. Let G be a finitely generated FC-group, Γ be its groupoid of adjointaction. Then X fin can be decomposed into a direct sum of submodules X fin (Γ) = M [ u ] ∈ G G X fin (Γ [ u ] ) . (12)We identify character supported on Γ [ u ] with its extension on the whole Γ ,such that the extended character has zero value on any morphisms outside of Γ [ u ] . Proof.
1) The inclusion L [ u ] ∈ G G X fin (Γ [ u ] ) ⊂ X fin (Γ) is obvious.In fact, if χ ∈ L [ u ] ∈ G G X fin (Γ [ u ] ) , then by definition of infinite direct sumof modules, χ is a finite sum of locally-finite characters. This implies that χ islocally-finite.2) Let us prove that X fin (Γ) ⊂ L [ u ] ∈ G G X fin (Γ [ u ] ) . This is actually equiv-alent to the following statement: if χ is locally finite, then supp ( χ ) lies in afinite union of subgroupoids F mi =1 Γ [ u i ] for some m . So, let us prove this.By assumption, G is finitely generated. Let us choose and fix its presentation: G = h s , . . . , s n | R i ≡ h S | R i . This means that any element v can be (perhaps non-uniquely) represented inthe form (see [35]): v = a . . . a k , where a , . . . , a k are generating elements of G or their inverses a = s ± i , . . . , a k = s ± i k . In terms of groupoid of adjoint action, it means that any morphism ( u, v ) =( u, a . . . a k ) ∈ Hom (Γ) can be decomposed into a composition of morphismsin the following way ( u, a . . . a k ) = ( u, a ) ◦ ( a − u, a ) ◦ . . . ◦◦ ( a − k − . . . a − u, a k − ) ◦ ( a − k − . . . a − u, a k ) . (13)Now consider a locally-finite character χ ∈ X fin (Γ) . Since χ is locally-finite,then for any generating element s i there is only a finite number of morphisms ofthe form ( h, s i ) , such that χ does not vanish on these morphisms (similarly to[3]). Hence there can only be a finite number of subgroupoids Γ [ u i ] containingmorphisms of the form ( h, s i ) , such that χ (( h, s i )) = 0 .Similarly for any generating element’s inverse s − i .Since G is finitely generated, all morphisms ( ∗ , g ) such that g ∈ S ⊔ S − and χ (( ∗ , g )) = 0 are contained in a finite union of a finite unions of subgroupoids Γ [ u i ] . This implies that all such morphisms are contained in a finite union ofsubgroupoids, and without any loss of generality we can denote it as F mi =1 Γ [ u i ] .14ow recall that any morphism ( u, v ) can be decomposed in the form (13). Weconclude that if χ (( u, v )) = 0 , then ( u, v ) ∈ Hom ( F mi =1 Γ [ u i ] ) . Hence supportof χ lies in F mi =1 Γ [ u i ] . Corollary 3.0.1.
The following decomposition holds X fin (Γ) = M [ u ] ∈ G G X fin (Γ [ u ] ) . (14) Proof.
The proof for this statement is completely the same as for the statementabove.
Theorem 3.1. If G is a finitely generated FC-group, all weakly-inner deriva-tions of A [ G ] are inner. In order to prove this theorem, we need the following lemma.
Lemma 3.2. If | [ u ] | < ∞ , then all weakly inner derivations whose charactersare supported on Γ [ u ] are inner derivations.Proof. If we apply lemma (2.1), we know that any trivial-on-loops character χ on Γ [ u ] can be expressed as a (possibly infinite) sum of adjoint derivationcharacters χ = X g ∈ G C g χ ad g . However, if supp ( C g χ ad g ) ⊂ Γ [ u ] , then g ∈ Ob (Γ [ u ] ) = [ u ] . It follows that χ = X g ∈ [ u ] C g χ ad g . Since [ u ] is finite, χ is a finite sum of adjoints. Hence it is a character of aninner derivation. Proof. (Theorem (3.1) ) Let d be a weakly inner derivation. Then its character χ d is locally-finiteand trivial on loops. Due to previous lemma (3.0.1), χ d lies in a direct sum X fin (Γ [ u ] ) . Therefore, it is a finite sum of characters, whose supports lie inparticular subgroupoids. It follows that χ d is a finite sum of characters, eachof those corresponding to a finite sum of adjoints. Thus, χ d is a finite sum offinite sums of adjoint derivations’ characters, and its corresponding derivation d is a finite sum of adjoints. This means that d is an inner derivation.Now we can derive the structure theorem for derivations of finite group ringsover finitely generated FC-groups. 15 heorem 3.2. Let G be a finitely generated FC-group, A be a unital commu-tative ring. Then Der ( A [ G ]) ∼ = Inn ( A [ G ]) ⊕ M [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) . (15) Proof. As A -modules (not as Lie algebras): Der ( A [ G ]) ∼ = X fin (Γ) . Following the lemma (3.1), X fin (Γ) = M [ u ] ∈ G G X fin (Γ [ u ] ) Following the theorem (2.4), we can combine the decomposition of charactermodules on subgroupoids (2.2.2) X fin (Γ [ u ] ) ∼ = X fin (Γ [ u ] ) ⊕ X fin ( Hom ( u, u )) From the corollary of lemma (2.3), we know that all characters on Γ [ u ] arelocally-finite. In addition, we know that there is an explicit desription of char-acters on loop hom-sets (2.5). Therefore, we have: X fin (Γ [ u ] ) ∼ = X fin (Γ [ u ] ) ⊕ Hom Ab ( Z ( u ) , A ) Now recall that we have an A -module isomorphism between Inn ∗ and X fin .If we combine it with corollary (3.0.1), we obtain the isomorphism: Inn ∗ ∼ = X fin = M [ u ] ∈ G G X fin (Γ [ u ] ) . Now recall that due to the theorem (3.1), we have
Inn ∗ = Inn . Regroupingterms in the equation above, we obtain the theorem claim:
Der ( A [ G ]) ∼ = X fin (Γ) = M [ u ] ∈ G G X fin (Γ [ u ] ) ∼ = ∼ = M [ u ] ∈ G G (cid:0) X fin (Γ [ u ] ) ⊕ Hom Ab ( Z ( u ) , A ) (cid:1) ∼ = Inn ∗ ( A [ G ]) M [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) ∼ = ∼ = Inn ( A [ G ]) M [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) . (16)In the next section, we will be interested in the case of finite groups. So thefollowing corollary will be important. Corollary 3.2.1.
In case G is finite, Inn ∗ ( A [ G ]) = Inn ( A [ G ]) .Proof. This is a special case of theorem (3.1).16
Finite group case
In this section group G is assumed to be finite. In this case we see that itsgroupoid of adjoint action Γ has both finite number of objects and morphisms.Thus, any character has finite support, so any character on Γ is a locally-finitecharacter.Thus, taking in account the corollary 3.2.1 we get Proposition 4.1. Out ∗ ( A [ G ]) ∼ = Out ( A [ G ]) . Let us refine the theorem 3.2 for the case of finite group.
Theorem 4.1.
There is a way to describe all derivations of A [ G ] in case G isfinite:1. Der ( A [ G ]) ∼ = Inn ( A [ G ]) ⊕ (cid:0) L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) (cid:1) , Out ( A [ G ]) ∼ = L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) . Notice that these isomorphisms are isomorphims of A -modules, and they are notcanonical.Proof.
1. As A -modules (not as Lie algebras): Der ∼ = X fin (Γ) . Due to finiteness of G : X fin (Γ) = X (Γ) . As an A -module of A -valued functions over a finite disjoint union of domains, X (Γ) can be decomposed: X (Γ) ∼ = M [ u ] ∈ G G X (Γ [ u ] ) . Now, due to the theorem (2.4) X (Γ [ u ] ) = X fin (Γ [ u ] ) ∼ = X fin (Γ [ u ] ) ⊕ X fin ( Hom ( u, u )) ≡≡ X (Γ [ u ] ) ⊕ X ( Hom ( u, u )) . Hence X (Γ) ∼ = M [ u ] ∈ G G (cid:0) X (Γ [ u ] ) ⊕ X ( Hom ( u, u )) (cid:1) . On one hand, we know that M [ u ] ∈ G G X (Γ [ u ] ) = X (Γ) ∼ = Inn .
17n the other hand,
Hom ( u, u ) is a group with respect to morphism composition,and this group is isomorphic to centralizer Z ( u ) . In other words, characters onfinite Hom ( u, u ) are nothing but additive homomorphisms from Z ( u ) to A . I.e.we can delineate the following A -module homomorphism X ( Hom ( u, u )) ∼ = Hom Ab ( Z ( u ) , A ) . Regrouping terms in the equation above, we obtain that
Der ∼ = X (Γ) ∼ = Inn ⊕ M [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) .
2. Trivially follows from the first part and proposition 4.1.
If the group G is finite, we can suppose that the dimension of the space of innerderivations is the difference between number of generators ad g , g ∈ G and thedimension of centralizer of a group algebra A [ G ] , which is equal to the number ofconjugacy classes | G G | ( here is the proof for the case A = C ). In this paragraphwe prove this formula for arbitrary A : dim ( Inn ( A [ G ])) = | G | − | G G | . In order to prove this, we need the following well-known fact (see [13], chapter2). For a fixed conjugacy class [ u i ] ∈ G G , denote K i := P g ∈ [ u i ] g . Statement.
An element x lies in the center of A [ G ] if and only if x = C K + . . . + C n K n , for some elements C i ∈ A . Theorem 4.2.
Inner derivations can be described in terms of an A -modulepresentation Inn = h ad g , g ∈ G | X g ∈ [ u ] ad g = 0 for any [ u ] ∈ G G i Details on module presentations can be found at [24]. This theorem is equiv-alent to the following
Corollary 4.2.1.
Let us choose some u , . . . , u n , such that all the u i ’s belongto different conjugacy classes. Then Inn ( A [ G ]) is a free A -module, with thefollowing basis: (cid:8) ad g | g ∈ G \ { u , . . . , u n } (cid:9) . So let’s prove the theorem with the corollary.18 roof.
Inn , we know that ad g , g ∈ G is a generatingset for Inn . In addition, we know that ad x = 0 ⇐⇒ x lies in the center ofthe group algebra A [ G ] . Combining this with the description of the center of agroup algebra, we obtain the claim of the theorem.In order to prove the corollary, it suffices to prove the rest for any conjugacyclass.So, we will fix a conjugacy class [ u ] = { g , . . . , g k } . Consider inner deriva-tions, such that their characters are supported in Γ [ u ] . In addition, let m ∈ { , . . . , k } .
2) Firstly, we have to prove that { ad g | g ∈ [ u ] \ g m } is truly a generating setfor inner derivations supported on Γ [ u ] .Denote Inn (Γ [ u ] ) := { d ∈ Inn ( A [ G ]) | supp ( χ d ) ⊂ Γ [ u ] } . Any inner derivation d supported in Γ [ u ] is a linear combination of adjoints ad g , such that g ∈ [ u ] . In addition, due to the lemma (4.1), X i g i ∈ C A [ G ] ( A [ G ]) = ⇒ ad P i g i = X i ad g i . Thus, ad g m is a linear combination of others: ad g m = X i,i = m − ad g i . X i,i = m C i ad g i = 0 for some non-zero coefficients C i ∈ A . Then for any x ∈ A [ G ] , we have X i,i = m C i ad g i )( x ) = X i,i = m C i ad g i ( x ) = X i,i = m C i [ g i , x ] = [ X i,i = m C i g i , x ] . (17)This means that this linear combination must lie in the group ring’s center: X i,i = m C i g i ∈ C A [ G ] ( A [ G ]) . We see that this contradicts (4.1). Therefore, all C i ’s must vanish, and { ad g | g ∈ [ u ] \ g m } must be linearly independent.4) Due to the lemma above((3.0.1)), Inn ( A [ G ]) = M [ u ] ∈ G G Inn (Γ [ u ] ) .
19s a consequence,
Inn ( A [ G ]) is a free finitely generated module, spanned by G i { ad g | g ∈ [ u i ] \ u i } == { ad g | g ∈ G \ { u , . . . , u n | u i ’s belong to different conjugacy classes }} . (18) Corollary 4.2.2. If A is a field, then dim ( Inn ) = n X i =1 ( | [ u i ] | −
1) = | G | − | G G | . Proof.
It’s a widely known fact that a free finitely generated module over a fieldis a vector space. Its dimension is equal to the number of basis elements, whichwe have calculated in the theorem (4.2.1).
Now we will describe characters corresponding to the loop groups. This givesus criterion to find out whether all derivations are inner.
Lemma 4.1.
Let H be any subgroup of G , A be a ring. If ϕ ∈ Hom Ab ( H, A ) ,then for any ∀ g ∈ H we have ord ( g ) ϕ ( g ) = 0 ∈ A Proof.
Firstly, if ord ( g ) = k in G , then g has the same order in H .Secondly, since | G | < ∞ , ∀ g ∈ H , we have ord ( g ) < ∞ . Then ϕ ( e ) = ϕ ( g ord ( g ) ) = ord ( g ) ϕ ( g ) . Corollary 4.2.3.
Let G be a finite group, A be a unital commutative ring.Then Der ( A [ G ]) = Inn ( A [ G ]) if and only if there exists [ u ] ∈ G G and a non-trivial homomorphism ϕ ∈ Hom Ab ( Z ( u ) , A ) such that ∀ g ∈ Z ( u ) , ord ( g ) ϕ ( g ) =0 ∈ A . Corollary 4.2.4. If A is a torsion-free ring, G is a finite group, then Out ( A [ G ]) =0 . As a consequence, if A = Z , Q , R , C , then Out ( A [ G ]) = 0 . Hereinafter we use notation gcd ( a, b ) for the greatest common divisor. Lemma 4.2.
Let A = Z m , G - finite. Then ∀ g ∈ G, gcd ( ord ( g ) , m ) = 1 = ⇒ Der ( A [ G ]) = Inn ( A [ G ]) . Proof.
Let ϕ ∈ Hom Ab ( Z ( u ) , Z m ) for some u . Then for any g ∈ Z ( u ) we have ord ( g ) ϕ ( g ) = 0 ∈ Z m = ⇒ m | ord ( g ) ϕ ( g ) . Since ∀ g ∈ G, gcd ( ord ( g ) , m ) = 1 , we have m | ord ( g ) ϕ ( g ) = ⇒ m | ϕ ( g ) = ⇒ ϕ ( g ) = 0 ∈ Z m . .3 Additive homomorphisms The following fact is widely-known amongst group theorists.
Statement.
Let ϕ : G → A be an additive homomorphism from a non-abeliangroup G to abelian group A . Let us denote π : G → G/ [ G, G ] as the projectionmapping. Then there exists ˆ ϕ , such that ϕ = ˆ ϕ ◦ π . G AG/ [ G, G ] ϕπ ˆ ϕ If we use the primary decomposition of finite abelian groups, then we candecompose the additive group of ring AA ∼ = Z p i ⊕ ... ⊕ Z p inn , and G/ [ G, G ] ∼ = Z q j ⊕ ... ⊕ Z q jmm . Lemma 4.3.
There is a nontrivial homomorphism ϕ : Z p i → Z q j if and only if p = q .Proof.
1. If p = q , then we can always take ϕ : 1 q j − .2. Let ϕ : Z p i → Z q j be a homomorphism. Notice that elements of theprimary group Z p i can only be of order , p, p , ..., p i due to the Lagrange theo-rem (order of element should divide order of group). Similarly we can say thatelements of Z q j can only be of order , q, ..., q j . Then for any element g ∈ Z p i of order p k we have ϕ (0) = ϕ ( p k g ) = p k ϕ ( g ) . On the other hand we knowthat ord ( ϕ ( g )) ∈ , q, ..., q j . It can only be possible if p = q . Corollary 4.2.5.
Let the primary decompositions of A and G/ [ G, G ] be A ∼ = Z p i ⊕ ... ⊕ Z p inn , G/ [ G, G ] ∼ = Z q j ⊕ ... ⊕ Z q jmm . Then there is a nontrivial homomorphism in ϕ ∈ Hom Ab ( G/ [ G, G ] , A ) if andonly if there is some p i , q j such that p i = q j . Now we can obtain the criterion for the triviality of the outer derivationsalgebra.Let the ring A be such that A ∼ = Z p i ⊕ ... ⊕ Z p inn is the primary decomposition for additive group of the ring A . Also let G/ [ G, G ] ∼ = Z q j ⊕ ... ⊕ Z q jmm be the primary decomposition of G/ [ G, G ] .21 heorem 4.3. For finite group G and finite ring A all derivations Der ( A [ G ]) are inner ( Der ( A [ G ]) = Inn ( A [ G ]) ) if and only if { p , ..., p n }∩{ q , ..., q m } = ∅ .Proof. { p , ..., p n } ∩ { q , ..., q m } = ∅ . Due to corollary (4.2.5) we knowthat this implies Hom ( G/ [ G, G ] , A ) = 0 . We know that for any homomorphism ϕ : G → A there is ˆ ϕ ∈ Hom ( G/ [ G, G ] , A ) such that ϕ = ˆ ϕ ◦ π . Thus ϕ = 0 ◦ π must be the zero homomorphism.2) Let { p , ..., p n } ∩ { q , ..., q m } = r. Without any loss of generality we can say that r = p k = q k . If it’s not we canjust reorder primary components. Then we have a nontrivial homomorphism ϕ k = Z r ik → Z r jk , ϕ k : 1 r j k − between primary subgroups Z r ik ⊂ G/ [ G, G ] and Z r jk ⊂ A correspondingly.Firstly, we can extend ϕ k onto the whole G/ [ G, G ] by zeros on all primarycomponents except for Z r ik . Thus we have obtained ˆ ϕ : G/ [ G, G ] → A .Secondly, we can just compose ˆ ϕ with the projection map π : ϕ = ˆ ϕ ◦ π. Now we can clearly see that there is a nontrivial central derivation d eϕ . Sincecentral derivations are isomorphic to a certain subalgebra of Out ( A [ G ]) , thenin this case Out ( A [ G ]) = 0 . In the previous paragraphs we built two isomorphisms of A -modules: "outerderivations" ↔ "characters on loops" and "characters on loops" ↔ "additivehomomorphisms on centralizers". Now we will provide the explicit descriptionof the composition of these isomorphisms. Problem is that such isomorphism isnot canonical, because it depends on choice of elements in conjugacy classes. Inthis paragraph we build one such A -module isomorphism explicitly.Fix conjugacy classes of G as G G = { [ u ] , [ u ] , . . . , [ u n ] } . Choose representatives of conjugacy classes: u ∈ [ u ] , . . . , u n ∈ [ u n ] . Denote F u ,...,u n : n M i =1 Hom Ab ( Z ( u i ) , A ) ∼ −→ Out ( A [ G ]) as the A -module isomorphism we built above. Let ϕ ∈ M [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) . heorem 5.1. Let ϕ ∈ Hom Ab ( Z ( u i ) , A ) for some i . The A -module isomor-phism mapping loop groups to outer derivations is described by the followingformula: F u ,...,u n ( ϕ ) = D ( ϕ ) + Inn ( D ( ϕ )) uv = ( ϕ ( gvg − ) if v − u = uv − = g − u i g for some g ∈ G , else . The isomorphism F u ,...,u n ( ϕ )) is then extended on L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) by linearity.Proof. If ϕ ∈ L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) , then ϕ is a direct sum of homomor-phisms: ϕ = ( ϕ , . . . , ϕ n ) , ϕ i ∈ Hom Ab ( Z ( u i ) , A ) . We have an outer derivation F u ,...,u n ( ϕ ) as an outer derivation, and we wishto calculate its coefficients. Then ( F u ,...,u n ( ϕ )) uv = ( n X i χ i )(( u, v )) = assume that ( u, v ) ∈ Γ [ u i ] = χ i (( u, v )) = ( χ ϕ i ( ψ ) , ψ is a loop, conjugated to ( u, v )0 , else (19)If ψ is a loop conjugated to ( u, v ) , then we have χ ϕ i ( ψ ) = χ ϕ i (( u, v )) . Since ( u, v ) is a loop in Γ [ u i ] , we know that exists g such that v − u = uv − = g − u i g .Let us find such a ψ : u i g − u i g ψ ( g − u i ,g − )( u i g,g ) ( u,v ) Since the character χ ϕ i is potential, χ ϕ i ( ψ ) does not depend on the choice of g .Hence we have: χ ϕ i (( u, v )) = χ ϕ i ( ψ ) = χ ϕ i (( u i g, g ) ◦ ( u, v ) ◦ ( g − u i , g − )) == χ ϕ i (( u i g, g ) ◦ ( vg − u i , vg − )) = χ ϕ i (( gvg − u i , gvg − )) == ϕ i ( gvg − ) . Combining this formula with the formula (19), we obtain the statement ofthe theorem. 23s a result, we can describe outer derivations as an A -module presentation.(Details on module presentations can be found at [24]). Denote the gener-ating set of Out as S , and relations as T . Its relations are inherited from L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) (just usual relations for addition and scalar multipli-cation on Hom -sets with A -module structure). In terms of the theorem above, Out = < S | T >, where S = G [ u ] ∈ G G S u Here S u = { F u ,...,u n ( ϕ ) | ϕ ∈ Hom Ab ( Z ( u ) , A ) } T = { F u ,...,u n ( ϕ ) + F u ,...,u n ( ψ ) = F u ,...,u n ( ϕ + ψ ) ,F u ,...,u n ( aϕ ) = aF u ,...,u n ( ϕ ) } for any a ∈ A and ϕ, ψ ∈ M [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) (20)Obviously, sets { S u | [ u ] ∈ G G } are disjoint. Z [ S ] Let us consider the following case: A = Z , G = S .In order to describe outer derivations, we need to calculate L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) and then map it to Out ( Z ( S )) by the means of theorem (5.1). Thus wecan calculate explicit form of outer derivations’ elements. So we begin with L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) .There are three conjugacy classes in S : [ e ] , [(12)] , [(123)] .For any permutation σ its centralizer is defined as Z ( σ ) = { g ∈ G | σ = g − σg } . Since centralizer is a subgroup, due to Lagrange’s theorem on subgroupindices it must be of size 1,2,3, or 6. On the other hand, conjugation leaves thecycle type intact. Therefore,conjugacy class [ u ] [e] [(12)] [(123)]centralizer Z ( u ) up to isomorphism S Z Z Now we have to describe all the additive homomorphisms mapping these cen-tralizers to Z . Proposition 5.1.
Additive hom-sets have the following form:1.
Hom Ab ( S , Z ) = { , ϕ : ( ij ) for ∀ ( ij ) } , Hom Ab ( Z , Z ) = { , ϕ : 1 } , Hom Ab ( Z , Z ) = { } . Proof.
The second and the third parts are obvious, because image of any elementin the power of its order should be equal to zero.24ow let us prove the first part of our statement. Consider an arbitraryadditive homomorphism ϕ ∈ Hom Ab ( S , Z ) .For any element g of S we need ord ( g ) ϕ ( g ) = 0 ∈ Z to hold true. So, only (12) , (13) , (23) can be mapped by ϕ to nonzero.Without any loss of generality we can assume that ϕ (12) = 2 . Then ϕ (23) + ϕ (13) = 2 = 2 ϕ (13) + ϕ (23) = ⇒ ϕ (23) − ϕ (13) = 0 Now if ϕ (23) = ϕ (13) = 0 , then ϕ (123) = ϕ (23) + ϕ (12) = 2 . This condradictsto the fact that ord (123) = 3 : ϕ ((123) ) = ϕ ( e ) = 0 = 2 Hence ϕ (23) = ϕ (13) = 2 , and the only nontrivial homomorphism in Hom Ab ( S , Z ) is the ϕ : ( ij ) for ∀ ( ij ) . We denote it as ϕ .Now we can use F e, (12) , (123) from the theorem (5.1) to map non-trivial ho-momorphisms contained in L [ u ] ∈ G G Hom Ab ( Z ( u ) , A ) to derivations.1) We begin with the nontrivial homomorphism ϕ from the first hom-set Hom Ab ( Z ( e ) , Z ) ∼ = Hom Ab ( S , Z ) . Denote the corresponding outer deriva-tion as d + Inn = F e, (12) , (123) ( ϕ ) . The theorem (5.1) gives us ( d ) uv = ( F e, (12) , (123) ( ϕ )) uv = = ( ϕ ( gvg − ) if v − u = uv − = e , else . So, nontrivial coefficients occur only when v − u = uv − = e = ⇒ u = v ,Moreover, we see that depending on the cycle type of u , by theorem (5.1) wehave ( d ) uu = , if u = e,ϕ ( u ) = 2 if u = ( ij )0 , if u = ( ijk ) . To conclude, ( d ) uv = ( if u = v is a transposition, , else . (21)2) The second nontrivial homomorphism is ϕ ∈ Hom Ab ( Z ((12)) , Z ) De-note the corresponding outer derivation as d + Inn = F e, (12) , (123) ( ϕ ) . ( d ) uv = ( F e, (12) , (123) ( ϕ )) uv == ( ϕ ( gvg − ) if v − u = uv − = g − (12) g for some g ∈ G , else . (22)On one hand, if v − u = uv − , then there can be 3 cases:1. either u = ( ij ) , v = e ,2. or u = e, v = ( ij ) ,3. or u, v are cycles of length 3.Since a product of two cycles of length 3 cannot be a transposition, third caseleads to a contradiction. On the other hand, we know that for any derivation d , it holds true that d ge = 0 for any g . Now we see that the first case leads to acontradiction.It follows that only one case is possible: u = e, v = ( ij ) . Now since g − (12) g can only be a transposition, it suffices to consider cases when g equalsto e, (13) , (23) . Therefore, equation (22) takes form: ( d ) uv == ϕ (12) = 2 if u = e, v = (12) , g = e,ϕ ( gvg − ) = ϕ ((13)(23)(13)) = ϕ ((12)) = 2 if u = e, v = (23) , g = (13) ,ϕ ( gvg − ) = ϕ ((23)(13)(23)) = ϕ ((12)) = 2 if u = e, v = (13) , g = (23) , , else == ( , u = e, v ∈ { (12) , (23) , (31) } , , else (23)Finally, the theorem (5.1) tells us that the Z -module of outer derivationsadmits the description in terms of generators and relations: Out = < S | T >, with the generating set equal to S = { d + Inn , d + Inn } and the relations are T = { d + Inn ) =
Inn = 2( d + Inn ) } . entral derivations of Z [ S ] Let z ∈ Z ( G ) , τ : G → A - additive homomorphism.Linear operator d zτ defined on generators as d zτ : x τ ( x ) xz , is a centralderivation (see [4]). Any linear combination of such operators is also a centralderivation. Proposition 5.2.
Central derivations are not isomorphic to outer derivations.Proof.
In our case, Z ( S ) = { e } = ⇒ only operators such as d eτ : x τ ( x ) xe = τ ( x ) x span ZDer . The corresponding outer derivation is d eτ + Inn .We can see that there is no such homomorphism τ ,that d eτ + Inn = d + Inn ,because characters of d eτ and d are not trivial on loops, but have supports indifferent subgroupoids. F m D n Here the F m stands for a finite field of order m .The main goal of this section is to reproduce the result of paper [19]. Inorder to calculate derivations of F m D n , we use several properties of dihedralgroups, proofs of which can be found in expository papers by Keith Conrad[17, 18].To begin, the dihedral group of size n can be described by its presentation: D n = h r, s | r n = s = ( rs ) = 1 i The following theorem is proved in [17], (see Theorem 4.1).
Theorem 5.2.
The group D n has • two conjugacy classes of size 1: { } , { r n } , • n-1 conjugacy classes of size 2: { r, r − } , { r , r − } , . . . , { r n − , r − n +1 } , • two conjugacy classes of size n: { r i s | ≤ i ≤ n − } , { r i +1 s | ≤ i ≤ n − } . As a result, there are n + 3 conjugacy classes.Let us choose and fix first element in any of conjugacy classes written above.Now we need to compute centralizers. Proposition 5.3. In D n we get the following centralizers:1. Z ( r i ) = { , r, . . . , r n − } , ≤ i ≤ n, . Z ( s ) = { , s } , Z ( r i s ) = { , r i s } . Proof.
The proposition is proved by direct computationNow we are able to describe additive homomorphisms from centralizers to F m .1. Hom Ab ( Z ( r i ) , F m ) = Hom Ab ( Z n , F m ) , ≤ i ≤ n ;2. Hom Ab ( Z ( s )) = Hom Ab ( Z , F m ) ;3. Hom Ab ( Z ( r i s ) , F m ) = Hom Ab ( Z , F m ) . Corollary 5.2.1.
Using the decomposition theorems above,1.
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