aa r X i v : . [ m a t h . G R ] F e b Maximality of reversible gate sets ⋆ Tim Boykett , , Institute for Algebra, Johannes-Kepler University, Linz, Austria Time’s Up Research, Linz, Austria [email protected]://timesup.org/cca University for Applied Arts, Vienna
Abstract.
In order to better understand the structure of closed collec-tions of reversible gates, we investigate the lattice of closed sets and themaximal members of this lattice. In this note, we find the maximal closedsets over a finite alphabet. We find that for odd sized alphabets, thereare a finite number of maximal closed sets, while for the even case wehave a countable infinity, almost all of which are related to an alternat-ing permutations. We then extend to other forms of closure for reversiblegates, ancilla and borrow closure. Here we find some structural results,including some examples of maximal closed sets.
For a given finite set A , we investigate the collections of reversible gates, or bijec-tions of A k for all k . In many senses, these are akin to a bijective, permutative orreversible clone. The work derived from Tomasso Toffoli’s work [12] and as suchwe call closed systems of bijections reversible Toffoli Algebras (RTAs). The workalso relates to permutation group theory, as an RTA C is a N -indexed collectionof permutations groups, C [ i ] ≤ Sym ( A i ).In previous papers, Emil Jeˇr´abek has found the dual structure [7] and theauthor, together with Jarkko Kari and Ville Salo, has investigated generatingsets [2,3] and other themes.In this paper, we determine the possibile maximal closed systems, relyingstrongly on Liebeck, Praeger and Saxl’s work[10], and determine some propertiesof maximal borrow and ancilla closed RTAs.We show that the maximal RTAs are defined by an index that defines thesingle arity at which the RTA is not the full set of bijections. We then showthat for different indices and orders of A , only certain possibilities can arise. Forancilla and borrow closed RTAs we find that there is similarly an index belowwhich the maximal RTAs are full symmetry groups and above which they arenever full.We start by introducing the background properties of RTAs and some per-mutation group theory. The rest of the paper is an investigation of maximality,with the main result, Theorem 4 taking up the main body of this section. Wethen investigate properties of borrow and ancilla closed RTAs. ⋆ The research has been supported by Austrian Science Fund (FWF) research projectsAR561 and P29931. T. Boykett
In this section we will introduce the necessary terminology.Let A be a finite set. Sym ( A ) = S A is the set of permutations or bijections of A , Alt ( A ) the set of permutations of even parity. If A = { , . . . , n } we will write S n and A n . We write permutations in cycle notation and act from the right. Wewrite the action of a permutation g ∈ G ≤ Sym ( A ) on an element a ∈ A as a g .A subgroup G ≤ S A is transitive if for all a, b ∈ A there is a g ∈ G such that a g = b . We also say that G acts transitively on A . A subgroup G of S A acts imprimitively if there is a nontrivial equivalence relation ρ on A such that forall a, b ∈ A , for all g ∈ G , aρb ⇒ a g ρb g . If there is no such equivalence relation,then G acts primitively on A .Let G be a group of permutations of a set A . Let n ∈ N . Then the wreathproduct GwrS n is a group of permutations acting on A n . The elements of GwrS n are { ( g , . . . , g n , α ) | g i ∈ G, α ∈ S n } with action defined as follows:for ( a , . . . , a n ) ∈ A n , ( a , . . . , a n ) ( g ,...,g n ,α ) = ( a g α − , . . . , a g n α − n ).Let B n ( A ) = Sym ( A n ) and B ( A ) = S n ∈ N B n ( A ). We call B n ( A ) the setof n -ary reversible gates on A , B ( A ) the set of reversible gates. For α ∈ S n ,let π α ∈ B n ( A ) be defined by π α ( x , . . . , x n ) = ( x α − (1) , . . . , x α − ( n ) ). We callthis a wire permutation . Let Π = { π α | α ∈ S n , n ∈ N } . In the case that α isthe identity, we write i n = π α , the n -ary identity. Let f ∈ B n ( A ), g ∈ B m ( A ).Define the parallel composition as f ⊕ g ∈ B n + m ( A ) with ( f ⊕ g )( x , . . . , x n + m ) =( f ( x , . . . , x n ) , . . . , f n ( x , . . . , x n ) , g ( x n +1 , . . . , x n + m ) , . . . , g m ( x n +1 , . . . , x n + m )).For f, g ∈ B n ( A ) we can compose f • g in Sym ( A n ). If they have distinct aritieswe “pad” them with identity, for instance f ∈ B n ( A ) and g ∈ B m ( A ), n < m ,then define f • g = ( f ⊕ i m − n ) • g and we can thus serially compose all elementsof B ( A ).We call a subset C ⊆ B ( A ) that includes Π and is closed under ⊕ and • a reversible Toffoli algebra (RTA) based upon Toffoli’s original work [12]. Thesehave also been investigated as permutation clones [7] and with ideas from cat-egory theory [8]. If we do not insist upon the inclusion of Π , the get reversibleiterative algebras [3] in reference to Malcev and Post’s iterative algebras. Fora set F ⊆ B ( A ) we write h F i as the smallest RTA that includes F , the RTA generated by F .Let q be a prime power, GF ( q ) the field of order q , AGL n ( q ) the collectionof affine invertible maps of GF ( q ) n to itself. Let Aff( q ) = S n ∈ N AGL n ( q ) be theRTA of affine maps over A = GF ( q ).Let C be an RTA. We write C [ n ] = C ∩ B n ( A ) for the elements of C of arity n . We will occasionally write ( a , . . . , a n ) ∈ A n as a a . . . a n for clarity.We say that an RTA C ≤ B ( A ) is borrow closed if for all f ∈ B ( A ), f ⊕ i ∈ C implies that f ∈ C . We say that an RTA C ≤ B ( A ) is ancilla closed if for all f ∈ B n ( A ), g ∈ C [ n +1] with some a ∈ A such that for all x , . . . , x n ∈ A , forall i ∈ { , . . . , n } , f i ( x , . . . , x n ) = g i ( x , . . . , x n , a ) and g n +1 ( x , . . . , x n , a ) = a implies that f ∈ C . If an RTA is ancilla closed then it is borrow closed. aximality of reversible gate sets 3 In this section we introduce some results from permutation group theory that willbe of use. The maximal subgroups of permutation groups have been determined.
Theorem 1 ([10]).
Let n ∈ N . Then the maximal subgroups of S n are conjugateto one of the following G .1. (alternating) G = A n
2. (intransitive) G = S k × S m where k + m = n and k = m
3. (imprimitive) G = S m wrS k where n = mk , m, k >
4. (affine) G = AGL k ( p ) where n = p k , p a prime5. (diagonal) G = T k . ( Out ( T ) × S k ) where T is a nonabelian simple group, k > and n = | T | ( k −
6. (wreath) G = S m wrS k with n = m k , m ≥ , k >
7. (almost simple)
T ⊳ G ≤ Aut ( T ) , T = A n a nonabelian simple group, G acting primitively on A Moreover, all subgroups of these types are maximal when they do not lie in A n ,except for a list of known exceptions. It is worth noting that in the imprimitive case, we have an equivalence rela-tion with k equivalence clases of order m , the wreath product acts by reorderingthe equivalence classes, then acting an S m on each equivalence class. In thewreath case, the set A is a direct product on k copies of a set of order m , thewreath product acts by permuting indices and then acting as S m on each index. Lemma 1.
Let A be a set of even order and n ≥ . Then S A wrS n ≤ Alt ( A n ) .Proof. S A wrS n is generated by S A acting on the first coordinate of A n and S n acting on coordinates.The action of S A on A n is even because for each cycle in the first coordinate,the remaining n − | A | n − times, which is even, so the action of S A lies in Alt ( A n ). S n is generated by S n − and the involution ( n − n ). By the same argument,each cycle of the action occurs an even number of times, so the action of S n − and the involution (( n − n ) on A n lies in Alt ( A n ) so we are done. ⊓⊔ We have a similar inclusion for affineness.
Lemma 2.
For n ≥ , AGL n (2) ≤ Alt (2 n ) .Proof. AGL n (2) is generated by the permutation matrices { π (1 ,i ) | i = 2 , . . . , n } and the matrix (cid:20) (cid:21) ⊕ i n − . These bijections are even parity because they onlyact on two entries, thus have parity divible by 2 n − modulo 2 which is 0. ⊓⊔ Lemma 3.
Let A be a set of even order. Then S A wrS ≤ Alt ( A ) iff divides | A | .Proof. The same argument as above applies for S A . The action of S swaps | A | ( | A |− pairs. This is even iff 4 divides | A | . ⊓⊔ T. Boykett
In this section, we will determine the maximal RTAs on a finite set A .We have some nice generation results from other papers that will be useful. Theorem 2 ([2] Theorem 5.9).
Let A be odd. If B ( A ) , B ( A ) ⊆ C ⊆ B ( A ) ,then C = B ( A ) . Theorem 3 ([3] Theorem 20). If Alt ( A ) ⊆ C ⊆ B ( A ) then Alt ( A k ) ⊆ C for all k ≥ . The techniques used in the proof of Theorem 3 can be adapted to prove somefurther results.
Lemma 4.
Let | A | ≥ , then h B ( A ) , B ( A ) i is 3-transitive on A .Proof. Let A = { , , , . . . } . Let a, b, c ∈ A be distinct. We show that we canmap these to 111 , , ∈ A .Suppose a , b , c all distinct. Let α = ( a a a )( b b b )( c c c ) ∈ B ( A ). Let β = ( a a b b c c ∈ B ( A ). Then γ = ( π (23) • ( α ⊕ i ) • π (23) ) • ( i ⊕ β ) satisfies the requirements.Suppose a , b , c contains two values, wlog suppose a = b . Let d ∈ A −{ a , c } . Let δ = ( a a a d ) ∈ B ( A ). Let λ = π ((23) • ( δ ⊕ i ) • π (23) . Then λ will map a, b, c to the situation is the first case.Suppose a = b = c . Then one of a , b , c or a , b , c must contain atleast two values, wlog let a , b , c be so. Then π (13) will give us the first case ifit contains three values, the second case if it contains two cases. ⊓⊔ The two following results are only relevant for even A . Lemma 5.
Let | A | ≥ , B ( A ) , B ( A ) ⊂ C ≤ B ( A ) . Then Alt ( A ) ⊆ C [3] .Proof. For | A | = 4, the result is shown by calculation in GAP [6] that the ordersagree.For | A | = 5 the result follows from Theorem 2.Suppose | A | ≥ B ( A ) ⊆ C , we have all 1-controlled permutations of A in C . By [3] Lemma 18, with P ⊂ Alt ( A ) the set of all 3-cycles, we have all2-controlled 3-cycles in C . Thus (111 112 113) ∈ C . B ( A ) ∪ B ( A ) is 3-transitiveon A by Lemma 4, so we have all 3-cycles in C , so Alt ( A ) ⊆ C . ⊓⊔ We know that this is not true for A of order 2, where B ( A ) generates agroup of order 1344 in B ( A ), which is of index 15 in Alt ( A ) and is included inno other subgroup of B ( A ). However we find the following. Lemma 6.
Let | A | be even, B ( A ) , B ( A ) , B ( A ) ⊂ C ≤ B ( A ) . Then Alt ( A ) ⊆ C [4] . aximality of reversible gate sets 5 Proof.
For A of order 4 or more, we use the same techniques as in Lemma 5.For A of order 2, we calculate. We look at C [4] as a subgroup of S . The wirepermutations are generated by permutations (2 , , , , , , , , , , , , , , B ( A ) as a subgroup of B ( A ) acting on the indices2,3,4 is generated by (1 , , , , , , , , , , , , , ,
16) and (1 , , A , so Alt ( A ) ⊆ C [4] . ⊓⊔ We can now state our main theorem.
Theorem 4.
Let A be a finite set. Let M be a maximal sub RTA of B ( A ) . Then M [ i ] = B i ( A ) for exactly one i and M belongs to the following classes:1. i = 1 and M [1] is one of the classes in Theorem 1.2. i = 2 , | A | = 3 , and M [2] = AGL (3) (up to conjugacy)3. i = 2 , | A | ≥ is odd and M [2] = S A wrS i = 2 , | A | ≡ and M [2] = S A wrS i = 2 , | A | ≡ and M [2] = Alt ( A ) i = 2 , | A | ≡ and M [2] = T (3) . ( Out ( T ) × S ) where T is a finitenonabelian simple group, with | A | = | T | (up to conjugacy)7. i = 2 , | A | ≡ and M [2] is an almost simple group (up to conjugacy)8. i ≥ , | A | is even and M [ i ] = Alt ( A i ) Proof.
Suppose
M < B ( A ) with i = j natural numbers such that M [ i ] = B i ( A )and M [ j ] = B j ( A ). Wlog, i < j , let N = h M ∪ B j ( A ) i . Remember that composi-tions of mappings of arity at least j will also be of arity at least j , so N [ k ] = M [ k ] for all k < j . Then M < N because N contains all of B j ( A ) and N < B ( A )because N [ i ] = M [ i ] = B i ( A ). Thus M was not maximal, proving our first claim.For the rest of the proof, take M maximal with M [ i ] = B i ( A ). Then M [ i ] isa maximal subgroup of B i ( A ).Suppose i = 1. Then B ( A ) = S A and we are interested in the maximalsubgroups of S A . From Theorem 1 we know that these are in one of the 7classes.Suppose i ≥
2. Then S iA ≤ M [ i ] so M [ i ] is transitive on A i . As Π [ i ] ≤ M [ i ] we also know that S A wrS i ≤ M [ i ] . Assume M [ i ] acts imprimitively on A i withequivalence relation ρ . Let a, b ∈ A i , aρb with a i = b i . By the action of S A actingon the i th coordinate we obtain a ′ ρb ′ with a j = a ′ j and b j = b ′ j for all j = i . Bythe action of S i on coordinates we can move this inequality to any index. Thusby transitivity we can show that ρ = ( A n ) and is thus trivial, so our actioncannot be imprimitive.We now consider the cases of A odd and even separately.Suppose i ≥ | A | is odd. If i ≥ M [1] = B ( A ) and M [2] = B ( A ),so by Theorem 2 we have all of B ( A ) and thus M is not maximal, a contradiction.Thus we have i = 2. M [1] = B ( A ) = S A and π (1 2) ∈ M so M contains S A wrS . If | A | ≥ Sym ( A ) so M [2] must be precisely this. So the case of A order 3 is left. We want to know whichmaximal subgroups of Sym ( A ) contain S A wrS . There are 7 classes of maximalsubgroups, we deal with them in turn. T. Boykett – Since π (1 2) ∈ M is odd on A , M [2] Alt ( A ). – From the discussion above we know that M [2] is transitive and primitive on A , so the second and third cases do not apply. – The permutations in S can be written as affine maps in Z and π (1 2) canbe written as (cid:20) (cid:21) , the off diagonal 2 × Z , so S wrS embedsin the affine general linear group. Thus M [2] = AGL (3) is one possibility. – The diagonal case requires | T | k − = 9 for some nonabelian finite simplegroup T , a contradiction. – The wreath case requires 9 ≥ , a contradiction. – By [4] all G acting primitively on A with subgroups that are nonabelianfinite simple groups are subgroups of Alt ( A ), and we have odd elements in M , so this is a contradiction.Thus the only maximal subgroup is M [2] = AGL (3).Suppose i ≥ | A | is even. We know from Theorem 3 that for i > Alt ( A i ) from ∪ ≤ j
For A of odd order, there is a finite number of maximal RTAs. We are left with the possibility for A of order a multiple of 4, that the diagonalor almost simple case can arise. aximality of reversible gate sets 7 The diagonal case with A of order equal to the order of a finite simple non-abelian group start with order 60. The other possibility is that | A | = | T | forsome finite simple nonabelian group T . The only known result in this directionis in [11] where they show that symplectic groups Sp (4 , p ) where p is a certaintype of prime, now known as NSW primes, have square order. The first of thesegroups is of order (2 · · · ) corresponding to A of order (2 · · · ) = 11760.We note that the sporadic simple groups have order that always contains a primeto the power one, so they are not of square order. We know that the Alternatinggroup can never have order that is a square, as the highest prime less than n will occur exactly once in the order of the group. It might be possible that thereare other finite simple groups of square order. As far as we are aware, there havebeen no further results in this direction.Each of these possibilities is far beyond the expected useful arities for com-putational processes.The other case is to look at almost simple groups. In the above we sawthat all primitive actions of degree 4 are alternating. In order to carry one, wecan hope to use results about primitive permutation groups of prime power [5]and product of two prime power [9] degrees. With these we should be able todetermine almost simple examples up to degree 30. Once agin this would includeall examples of arities expected to be useful for computational processes. The strength of Theorem 4 is partially due to the fact that there is no effect ofthe existence of mappings of a certain arity in a given RTA on the size of thelower arity part, as there are no operators to lower the arity of a mapping. Thisdoes not apply with ancilla and borrow closure. In this section we collect someresults about maximal ancilla and borrow closed RTAs. The following resultreflects the first part of Theorem 4.
Lemma 7.
Let M ≤ B ( A ) be a maximal borrow or ancilla closed RTA. Thenthere exists some k ∈ N such that for all i < k , M [ i ] = B i ( A ) and for all i ≥ k , M [ i ] = B i ( A ) .Proof. Suppose M [ k ] = B k ( A ). Then for all f ∈ B m ( A ), m < k , f ⊕ i k − m ∈ M so f ∈ M , so M [ m ] = B m ( A ) for all m ≤ k . As M is maximal, there must be alargest k for which M [ k ] = B k ( A ), since otherwise M = B ( A ). ⊓⊔ From Theorem 2 we then note the following.
Lemma 8.
Let | A | be odd. Then k = 1 , are the only options in Lemma 7. In this case, we can say a bit more in case k = 2. If A is of order 3, then by theargument in Theorem 4 above, we find that M = Aff( A ), the affine maps over afield of order 3. Otherwise A is at least 5 and B ( A ) is no longer affine. See theresult below.Similarly, we obtain the following, but see Corollary 2 below. T. Boykett
Lemma 9.
Let | A | ≥ be even. Then in Lemma 7, k = 1 , , are the onlyoptions and for i > k , M [ i ] = Alt ( A i ) .Proof. We start by noting that for even | A | , for all f ∈ B i ( A ), f ⊕ i ∈ Alt ( A i +1 ).Thus if M [ i ] = Alt ( A i ) for some i > k , then M [ i − = B i ( A ) which is a contra-diction, which shows the second part of the result.Suppose k ≥
4, so B ( A ) , B ( A ) , B ( A ) ⊆ M . Then by Lemma 6 Alt ( A ) ⊆ M , so by Theorem 3 Alt ( A j ) ⊆ M for all j ≥
5. But we know that by borrowclosure, this implies that B j − ( A ) ⊆ M so M is in fact B ( A ). This is a contra-diction, so k < ⊓⊔ Using the same arguments, we obtain the following.
Lemma 10.
Let | A | = 2 . Then in Lemma 7, k = 1 , , are the only optionsand for i > k , M [ i ] = Alt ( A i ) .Proof. Suppose M is maximal with k ≥
5. Then by Theorem 3 we obtain M [ i ] = Alt ( A i ) for all i ≥
5, which by the first argument in the previous Lemma, impliesthat M is not maximal.Suppose M is maximal with k = 4. We know that M [3] = B ( A ). Then byLemma 6 we find that M [4] = Alt ( A ), so by Theorem 3 we obtain all of Alt ( A )so by borrow closure all of B ( A ) and thus M is not maximal. ⊓⊔ In general we obtain some examples of maximal borrow and ancilla closedRTA.
Lemma 11.
For | A | ≥ , the degenerate RTA Deg ( A ) generated by B ( A ) is amaximal borrow or ancilla closed RTA.Proof. Let
Deg ( A ) be generated by B ( A ) = S A . Then Deg ( A ) [ i ] = S A wrS i forall i ≥ B i ( A ) by Theorem 1. Thus any RTA N properlycontaining Deg ( A ) will have N [ i ] = B i ( A ) for some i ≥ N [2] = B ( A )by Lemma 7. Let f ∈ N [2] − Deg ( A ) [2] , then f ⊕ f ∈ N [4] − Deg ( A ) [4] so N [4] = B ( A ) and thus by Lemmas 8 and 9, N = B ( A ), so Deg ( A ) is maximal. ⊓⊔ Corollary 2.
Let | A | ≥ be even. Then in Lemma 7, k = 1 , are the onlyoptionsProof. From Lemma 9 we know k = 1 , , M is maximalin B(A) with k = 3.Suppose | A | = 4. B ( A ) can be embedded in B ( A ) represented on S with the tuples in A represented by the integers 1 , . . . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . . .. . . (241 , , , , , , , , , , , , , , , aximality of reversible gate sets 9 and (1 , , . . . (241 , S that is the alternating group, so M [4] = Alt ( A ) and by Theorem3 we then get M [5] = Alt ( A ) and thus M is not maximal.Suppose A is even with more than 6 elements. The degenerate RTA Deg ( A ) ≤ M because M [1] = B ( A ), but because Deg ( A ) is maximal and M [2] is a super-group of Deg ( A ) [2] , M is all of B ( A ) and is not maximal. ⊓⊔ Lemma 12.
Let A be of prime power order. Then Aff( A ) is a maximal borrowclosed RTA and a maximal ancilla closed RTA.Proof. Let M = Aff( A ). For every i , M [ i ] is maximal in B i ( A ) by Theorem 1.Suppose M is not maximal, so M < N < B ( A ).Let f ∈ B n ( A ), f ∈ N − M . Then N [ n ] = B n ( A ) by subgroup maximality,so for all i < n , N [ i ] = B i ( A ). For all j , f ⊕ i j ∈ ( N − M ) [ n + j ] so similarly N [ n + j ] = B n + j ( A ) so N = B ( A ) and M is maximal.Because Aff( A ) is ancilla closed and maximal as borrow closed, there can beno ancilla closed RTA between Aff( A ) and B ( A ) so Aff( A ) is a maximal ancillaclosed RTA. ⊓⊔ By [1] we know that for A of order 2, we have the following maximal ancillaclosed RTAs. – The affine mappings, – The odd prime-conservative mappings, that preserve the number of 1s mod p , an odd prime, – The parity respecting mappings, which either preserve the number of 1s mod2, or invert it.The affine mappings have k = 3 in Lemma 7 above, the parity respecting k = 2and the prime-conservative mappings k = 1.It remains open whether these are the borrow closed maximal RTAs over A of order 2.For A of order 3, we know that the affine maps are ancilla closed and as theyform a maximal borrow closed RTA, they also form a maximal ancilla closedRTA. Here k = 2 in the above. For A of order 4, we do not know which maximalRTA arise with k = 2. For A of order 5 or more, we know that k = 2 arises onlyfor the degenerate RTA Deg ( A ). We have determined the maximal RTAs, using results from permutation grouptheory and some generation results.As we have not been able to construct explicitly an example of a maximalRTA with i = 2 and M [ i ] of diagonal or almost simple type, the conjecture re-mains that these are not, in fact, possible. We note however that if such examplesexist, they will arise for A of order 8 or more, so will probably not be relevantfor any practical reversible computation implementation. In future work we aim to determine the weight functions as described by [7]for maximal RTAs, in order to determine whether they hold some interestinginsights.The results for borrow and ancilla closed RTAs are not as comprehensive, butwe see that it should be possible to determine these in the foreseeable future.For the ancilla case, we imagine that many of the techniques of [1] will proveuseful. In the ancilla case, we know all maximal RTA with k = 2 except for A oforder 4. One of the ongoing tasks is to investigate maximal ancilla closed RTAwith k = 1. Michael Guidici has helped extensively with understanding primitive permuta-tions groups, for which I thank him greatly.
References
1. Scott Aaronson, Daniel Grier, and Luke Schaeffer. The classification of reversiblebit operations.
Electronic Colloquium on Computational Complexity , (66), 2015.2. Tim Boykett. Closed systems of invertible maps.
Journal of Multiple-Valued Logicand Soft Computing , 32(5-6):565–605, 2019.3. Tim Boykett, Jarkko Kari, and Ville Salo. Finite generating sets for reversiblegate sets under general conservation laws.
Theor. Comput. Sci. , 701(C):27–39,November 2017.4. Francis Buekenhout and Dimitri Leemans. On the list of finite primitive permu-tation groups of degree ≤ Journal of Symbolic Computation , 22(2):215 – 225,1996.5. Qian Cai and Hua Zhang. A note on primitive permutation groups of prime powerdegree.
Journal of Discrete Mathematics , 2015.6. The GAP Group.
GAP – Groups, Algorithms, and Programming, Version 4.10.2 ,2019.7. Emil Jeˇr´abek. Galois connection for multiple-output operations.
Algebra Univer-salis , 79(2):Art. 17, 37, 2018.8. Y LaFont. Towards an algebraic theory of boolean circuits.
Journal of Pure andApplied Algebra , 184:257–310, 2003.9. Cai Heng Li and Xianhua Li. On permutation groups of degree a product of twoprime-powers.
Communications in Algebra , 42(11):4722–4743, 2014.10. Martin W Liebeck, Cheryl E Praeger, and Jan Saxl. A classification of the maximalsubgroups of the finite alternating and symmetric groups.
Journal of Algebra ,111(2):365 – 383, 1987.11. M. Newman, D. Shanks, and H. C. Williams. Simple groups of square order andan interesting sequence of primes.
Acta Arithmetica , 38(2):129–140, 1980.12. Tom Toffoli. Reversible computing. In
Automata, languages and programming(Proc. Seventh Internat. Colloq., Noordwijkerhout, 1980) , volume 85 of