A

solvable group has a subnormal series of subgroups.

5 and the smallest non

solvable group containing four conjugacy classes.

Recently, Guo and Shum pushed further this approach and obtained some charaterizations for a

solvable group and a p-

solvable group based on the assumption that some of its subgroups are CAP-subgroups (see [7]).

Abels:1971] Abels Herbert, 1971, An example of a finitely presented

solvable group, London Mathematical Society lecture notes, Cambridge: Cambridge University Press, pp.

cocompact lattice) of a direct product of a p-adic

solvable group with a connected, solvable Lie group.

3 shows that there exists a

solvable group [GAMMA] such that W([GAMMA]) contains S[L.

A long standing open problem in the character theory of finite

solvable groups is whether the derived length dl(G) of a

solvable group G is bounded above by the cardinality of cd(G), the set of irreducible character degrees of that group, i.

We begin by proving that the generalized free product of a nilpotent group and a

solvable group is not necessarily perfect.

Here we recall that every binary

solvable group except [Q.

And for a general connected

solvable group S, the Lie subalgebra t [intersection] [s, s] is always central in s.

Let G be a

solvable group in which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for 1 [less than or equal to] [[alpha].

Let G be a

solvable group, then one of the following holds.