Groups with small cardinality of the cubes of their two-elements subsets. (English) Zbl 0574.20021

Brailovsky, L. V.; Freĭman, G. A.

Collective phenomena, 5th Int. Conf., Moscow 1981, Ann. N.Y. Acad. Sci. 410, 75-82 (1983).

[For the entire collection see Zbl 0567.00002.]
Main results:
Theorem 1. Let \(| K^ 3| \leq 6\) for any two-element subset \(K\) of the finite group \(G\). Then one of the following possibilities holds: (1) \(G\) is abelian, (2) \(G=C_ 2\leftthreetimes H\) where \(| C_ 2| =2\) and \(axa=x^{-1}\) for \(a\in C_ 2^{\#}\), \(x\in H\); (3) \(\exp G/Z(G)=2\); (4) \(G\) is a 2-group, \(\exists a\in G\), \(a^ 2\not\in Z(G)\), \(a^ 4\in Z(G)\), \(a^ 8=1\) and \(G/<a^ 4>\) satisfies condition (2) of this theorem.
Theorem 3. Let \(| K^ 3| \leq 7\) for any two element subset \(K\) of the finite group \(G\). Then \(G\) satisfies one of the following conditions: (1) \(\exp G=3\), (2) \(| G:T| =2\) where \(T\in \mathrm{Syl}_ 3(G)\), \(T\) is non- abelian, and \(G/Z(G)\) is a Frobenius group, (3) \(G=T\leftthreetimes D\), \(D\in \mathrm{Syl}_ 2(G)\), \(T\in \mathrm{Syl}_ 3(G)\) and \(G/O_ 3(G)\) is a Frobenius group, or (4) \(G\) possesses a normal commutative 2-complement. Every group satisfying one of the conditions 1–3 also satisfies the condition \(| K^ 3| \leq 7\).

Reviewer: Ya. G. Berkovich

MSC:

20D60	Arithmetic and combinatorial problems involving abstract finite groups
20D40	Products of subgroups of abstract finite groups

Keywords:

two-element subset; finite group; Frobenius group; 2-complement

Citations:

Zbl 0567.00002