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Is it true that any abelian $p$-group (not necessarily finite) is the center of a $p$-group and of index $p^2$?

Is there an algorithm to determine all the Coxeter subgroups of a given Coxeter group? If we only want the Coxeter subgroups of finite index does that make the question easier? If we only want a Coxet …

In which classes of groups is it feasible to classify those groups in which the centralizer of every non-identity element is cyclic?

If G is a finite group which contains a maximal subgroup M which is abelian, then it is an exercise to show that G is solvable and that the third term in the derived series equals 1. What happens for …

Which finite simple groups have order N so that N+1 is a proper power? As an example: the simple group of order $168=13^2-1$.

Can one characterize the amenable groups that have the property that the centralizer of every non-identity element is cyclic? For example, must they be solvable?

From Wikipedia article on the Feit-Thompson Theorem proving a conjecture of Burnside that groups of odd order are solvable: "The attack on Burnside's conjecture was started by Michio Suzuki (1957), wh …

Can you show any finite subgroup of $SL_2(R)$ is cyclic without using an invariant form?

Suppose $G$ is a finitely generated subgroup of $GL_n(Z)$, $n\ge 3$. I suspect that there is no decision procedure for deciding whether or not such $G$ is finitely presented. How can this proved?

Which finite groups have uniqueness for the ordered sequence of composition factors (up to isomorphism)?