Clara Franchi

On Permutation Groups of Finite Type

A permutation group G is said to be a group of finite type { k }, k a positive integer, if each non-identity element of G has exactly k fixed points. We show that a group G can be faithfully represented as an irredundant permutation group of finite type if and only if G has a non-trivial normal partition such that each component has finite bounded index in its normalizer. An asymptotic structure theorem for locally (soluble-by-finite) groups of finite type is proved. Finite sharp irredundant permutation groups of finite type, notp -groups, are determined.

On minimal degrees of permutation representations of abelian quotients of finite groups

For a finite group G, we denote by μ(G) the minimum degree of a faithful permutation representation of G. We prove that if G is a finite p-group with an abelian maximal subgroup, then μ(G/G′) ≤ μ(G). 2010 Mathematics subject classification: primary 20B05.

Non-abelian sharp permutationp-groups

A permutation groupG of finite degreed is called a sharp permutation group of type {k},k a non-negative integer, if every non-identity element ofG hask fixed points and |G|=d−k. We characterize sharp non-abelianp-groups of type {k} for allk.

Min-wise independent groups

Let Ω be a finite set of cardinality n with a linear order on it and let k be a positive integer. Let F be a set of permutations on Ω and let PrF be an arbitrary distribution of probability on F The set F is said to be biased k-restricted min-wise independent if for every subset X of Ω such that |X| ≤ k, and every x ∈ X, when π is chosen at random in F, we have that PrF(min π(X) = π(x)) = /1|X|;. We study biased k-restricted min-wise independent groups. In particular, we prove that when k is close to n, then biased k-restricted min-wise independent groups are transitive.

MULTIPLE TRANSITIVITY AND MIN-WISE INDEPENDENCE IN PERMUTATION GROUPS

Let Ω be a finite linearly ordered set and let k be a positive integer. A permutation group G on Ω is called co-k-restricted min-wise independent on Ω if \[ \Pr(\min \pi(X)=\pi (x))=\frac{1}{|X|} \] for any X⊆Ω such that |X|≥|Ω|-k+1 and for any x∈X. We show that co-k-restricted min-wise independent groups are exactly the groups with the property that for each subset X⊆Ω with |X|≤k-1, the stabilizer G{X} of X in G is transitive on Ω\X. Using this fact, we determine all co-k-restricted min-wise independent groups.

A characterization of HN

Many of the sporadic simple groups appear to have (at least) two characteristics in the sense, for example, that there exist two prime divisors p and q of jGj such that mpðGÞd 2cmqðGÞ and, for r A fp; qg, every r-local overgroup M of a Sylow rsubgroup of G satisfies F ðMÞ 1⁄4 OrðMÞ. In contrast, hardly any other finite simple groups satisfy hypotheses of this type. Beginning with an unpublished manuscript of Gorenstein and Lyons, there has been an ongoing project to characterize the large sporadic groups (and some other simple groups) starting from hypotheses of this type. The Gorenstein–Lyons hypotheses include the assumption that q 1⁄4 2 and that m2;pðGÞd 3. This rapidly forces p 1⁄4 3. There are, however, several sporadic groups which should be regarded as groups of bicharacteristic fp; qg with fp; qg0 f2; 3g. A characterization of the Lyons group as a group of bicharacteristic f3; 5g appears in [4]. In this paper, we provide such a characterization of the Harada–Norton sporadic simple group as a group of bicharacteristic f2; 5g. Our notation and terminology is consistent with that introduced in [7]. In particular, by LpðGÞ, we mean the set of all quasisimple groups K such that K is a component of CGðxÞ=Op 0 ðCGðxÞÞ for some x A G of order p. Moreover, by ChevðpÞ we mean the set of all quasisimple groups K such that K=ZðKÞ is simple of Lie type in characteristic p. Finally, we call a finite group G K-local if every simple section of every local subgroup of G is a known finite simple group.

A note on groups paralyzing a subgroup series

We consider groups Γ of automorphisms of a groupG acting by means of power automorphisms on the factors of a normal series inG with lengthm. We show that [G, Γ] is nilpotent with class at mostm and that this bound is best possible. Moreover, such a Γ is parasoluble with paraheight at most 1/2m(m+3)+1, provided Γ′ is periodic. We give best possible bound in the case where the series is a central one.

m-Wielandt series in infinite groups

In a group G, u m ( G ) denotes the subgroup of the elements which normalize every subnormal subgroup of G with defect at most m . The m -Wielandt series of G is then defined in a natural way. G is said to have finite m -Wielandt length if it coincides with a term of its m -Wielandt series. We investigate the structure of infinite groups with finite m -Wielandt length.

Finite group actions on 3‐manifolds and cyclic branched covers of knots

We show that a hyperbolic

Radicals of

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