Volume 14, Issue 2

Let p be a rational prime. The k ( GV ) theorem states that, given a finite p ′-group G acting faithfully on a finite elementary abelian p -group V , the number of conjugacy classes of the semidirect product GV is bounded above by the order of V ( k ( GV ) ⩽ | V |). In the present paper we examine when the upper bound k ( GV ) = | V | is attained. It is shown that for p > 5 this happens if and only if G / C G ( U ) ≌ C G ( V / U ) is cyclic of order | U | – 1 for each nontrivial irreducible submodule U of V (Singer cycle). It remains open whether this is also true when p = 5. For p = 2, 3 there exist examples where equality holds but G is not abelian.

The Fong–Swan theorem showed that for each irreducible Brauer character ϕ of a finite solvable group, there exists at least one ordinary irreducible character χ that lifts ϕ . Recently there has been some progress toward understanding the set of lifts of a given Brauer character, by constructing certain canonical sets of lifts and studying their properties. In this paper we develop the theory of lifts of Brauer characters that are induced from inductive pairs. This leads to a lower bound on the number of ‘well-behaved’ lifts of a given Brauer character in terms of the inductive pair. Moreover, it is shown that given a ‘well-behaved’ lift, the vertices for the lift (a generalization of the vertices of the corresponding Brauer character) are unique up to conjugacy.

Based on the second author's thesis [Horn, Involutions of Kac–Moody groups, TU Darmstadt, 2008], in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac–Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established in [Helminck, Wang, Adv. Math. 99: 26–96, 1993] and [Springer, Algebraic groups and related topics: 525–543, North-Holland, 1984] for algebraic groups and in [Kac, Wang, Adv. Math. 92: 129–195, 1992] for certain Kac–Moody groups, we analyze the filtration studied in [Devillers, Mühlherr, Forum Math. 19: 955–970, 2007] in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised in [Bennett, Gramlich, Hoffman, Shpectorov, Curtis–Phan–Tits theory: 13–29, World Scientific, 2003].

For W a Coxeter group, let = { w ∈ W | w = xy where x , y ∈ W and x 2 = 1 = y 2 }. It is well known that if W is finite then W = . Suppose that w ∈ . Then the minimum value of ℓ( x ) + ℓ( y ) – ℓ( w ), where x , y ∈ W with w = xy and x 2 = 1 = y 2 , is called the excess of w (ℓ is the length function of W ). The main result established here is that w is always W -conjugate to an element with excess equal to zero.

By using tools from additive combinatorics, invariant theory and bounds on the size of the minimal generating sets of PSL 2 (𝔽 q ), we prove the following growth property. There exists ɛ > 0 such that the following holds for any finite field 𝔽 q . Let G be the group SL 2 (𝔽 q ), or PSL 2 (𝔽 q ), and let A be a generating set of G . Then | A · A · A | ⩾ min {| A | 1 + ɛ , | G |}. Our work extends the work of Helfgott [Helfgott, Ann. of Math. 167: 601–623, 2008] who proved similar results for the family {SL 2 (𝔽 p ): p prime}.

Let G = N wr H = N 2 n ⋊ H be the wreath product of N by H , where N is a finite nilpotent group, and H is a generalized quaternion group or a dihedral group of order 2 n . Then the normalizer property holds for G .

The paper deals with decomposition of a finite abelian group into a direct product of subsets. A family of subsets, the so-called uniquely complemented subsets, is singled out. It will be shown that if a finite abelian group is a direct product of uniquely complemented subsets, then at least one of the factors must be a subgroup. This generalizes Hajós's factorization theorem.