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In this paper we study the locus of generalized degree d Hénon maps in the parameter space RatdN of degree d rational maps PN →PN modulo the conjugation action of SLN+1. We show that Hénon maps are never in the GIT stable locus, that they are in the GIT unstable locus if d ≥ 3, and that they are in the strictly GIT semistable locus if N = d = 2. We also give a general classification of all unstable maps in Rat22. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let K be the conjugacy class of x ∈ G. If K² is a conjugacy class of G, then [x, G] is solvable. If the order of x is a power of prime, then [x, G] has a normal p-complement. We also prove some related results on the solvability of certain normal subgroups when a non-trivial coset has certain properties. [ABSTRACT FROM AUTHOR]

It is shown that if G is a group of even order with trivial center such that ΙGΙ > 2ΙCG(t)Ι3 for some involution t ∈ G, then there exists a proper subgroup H of G such that ΙGΙ < ΙHΙ2. If ΙGΙ > ΙCG(t)Ι3 and k(G) is the class number of G, then ΙGΙ ≤ k(G)3/Ι [ABSTRACT FROM AUTHOR]