‎in 1970‎, ‎menegazzo [gruppi nei quali ogni sottogruppo e intersezione di sottogruppi massimali‎, ‎ atti accad‎. ‎naz‎. ‎lincei rend‎. ‎cl‎. ‎sci‎. ‎fis‎. ‎mat‎. ‎natur. 48 (1970)‎, ‎559--562.] gave a complete description of the structure of soluble $im$-groups‎, ‎i.e.‎, ‎groups in which every subgroup can be obtained as intersection of maximal subgroups‎. ‎a group $g$ is said to have the $fm$...

A p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n− 1 is the dimens...

The finite ring Zk = Z(+, .) mod p k of residue arithmetic with odd prime power modulus is analysed. The cyclic group of units Gk in Zk(.) has order (p − 1).p , implying product structure Gk ≡ Ak.Bk with |Ak| = p− 1 and |Bk| = p , the ”core” and ”extension subgroup” of Gk respectively. It is shown that each subgroup S ⊃ 1 of core Ak has zero sum, and p+1 generates subgroup Bk of all n ≡ 1 mod p...

Glauberman’s classical Z∗-theorem is a theorem about involutions of finite groups (i.e. elements of order 2). It is one of the important ingredients for the classification of finite simple groups, which in turn allows to prove the corresponding theorem for elements of arbitrary prime order p. Let us recall the statement: if G is a finite group with a Sylow p-subgroup P , and if x is an element ...

Let F be a (topologically) finitely generated free pro-p-group, and ß an automorphism of F . If p ^ 2 and the order of ß is 2 , then there is some basis of F such that ß either fixes or inverts its elements. If p does not divide the order of ß , then the subgroup of F of all elements fixed by ß is (topologically) infinitely generated; however this is not always the case if p divides the order o...

Let G be a finite group and p an odd prime. By M < G we denote that M is a proper subgroup of G. Put the set Ψ p (G) = {M:M < G, |G : M| 6= a prime power and |G : M|p = 1}. In this paper we investigate the structure of G if every member of Ψ p (G) is nilpotent.

We present a heuristic asymptotic formula as x → ∞ for the number of isogeny classes of pairing-friendly elliptic curves over prime fields with fixed embedding degree k ≥ 3, with fixed discriminant, with rho-value bounded by a fixed ρ0 such that 1 < ρ0 < 2, and with prime subgroup order at most x.