By Stellmacher B.

Enable S be a finite non-trivial 2-group. it really is proven that there exists a nontrivial attribute subgroup W(S) in S satisfying:W(S) is common in H for each finite Σ4-free teams H withSεSyl2(H) andC H(O2(H))≤O2(H).

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**Extra info for A characteristic subgroup of Sigma4-free groups**

**Sample text**

The structures introduced here are derived from the pseudofields defined by Tits in [39] and form the near-domains of Karzel, and hence the name KT-field. Definition: A KT-field is a quadruple (F, +,·,a), which satisfies the following axioms: KT 1: (F, +, ·) is a near-domain KT2: a is an involutory automorphism of the multiplicative group (F*, ·) which satisfies the jUnctional equation a(1 + a(x)) = 1- a(l + x), for all x E F\{0, -1}. The characteristic of (F, +, ·, a) is defined to be the characteristic of (F, +, ·).

D), where d is a product of elements in D. But ord JlT/ ord (1, d 1, b) =p =? (1, d 1, b)P = id = <0, 1). l'. 1) (v) implies da,b =a- 1d 1,a-•b a = 1, for all a, b E F, a :f:: 0, and F lS a near-field. § 8. J = {1}. 1) (iii), (xi)). By definition E contains every sub near-field of F. The following two propostions involving the set E are proved in [28]. 16) For a, bEE and c E F, we have: (i) -a EE (ii) a- 1 E E, for a =I= 0 (iii) an (iv) c E E $>a+ (c +a) E E (v) (a+b)2EE. abE E. =? 18) Let (F, +, ·) be a near-domain with char F > 2.

10. Structure. , + . 1) The groups G0 form a class of maximal conjugate 'subgroups of G and each G0 operates sharply 2-transitively on M\ {a}. Further, define K = H 3 ={a E G: ord 0! = 3}, and let J and 11 = H2 denote the set of involutions and the set of involutions which have fixed points respectively. Then from § 2, K and ft are conjugate classes in G and G is of type (m, n), mE{0,1,2}, nE {0, 1} if and only if the elements of K have m, and the elements of ft have n + 1 fixed points. Define 10 = J n G0 = ft n G0 , for a EM.

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