Converse of Lagrange Theorem for p-groups.

Question

I need clarification on my understanding of Sylow Theorems.

Can I say that a finite p-group will have a subgroup for each prime power? If the above is valid, can I say then that p-groups satisfy the converse of Lagrange Theorem?

I'm not sure of that statement. Can someone show me a counterexample of it if not? Thanks

Yes the converse of Lagrange's theorem holds for finite $p$-groups. — Derek Holt, Jun 27 '16 at 09:01

score 1 · Accepted Answer · answered Jun 27 '16 at 09:27

The question of Alnitak refers to the class of finite groups known as CLT groups, where CLT stands for Converse Lagrange Theorem:

$G$ is a CLT group if for each positive integer $d$ dividing $|G|$, $G$ has at least one subgroup of order $d$.

These groups have been studied extensively. It turns out for example that all supersolvable groups are CLT, and all CLT groups are solvable. See also one of the early papers of Henry G. Bray, Pac. J. Math 27 (1968). All $p$-groups, or in general nilpotent groups, are supersolvable.

Converse of Lagrange Theorem for p-groups.

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