Converse of Lagrange's Theorem

Asked Feb 17 '15 at 15:09

Active Feb 17 '15 at 16:19

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I want to know partial converse of Lagrange's theorem is true upto how much ?

We know that it holds only in case of cyclic groups.

Also if a group has order $p^m\times n;\gcd(m,n)=1$ then it has a subgroup of order $p^i;1\leq i\leq m$ by sylow's theorem

But can we say something about existence of a subgroup of order $pm$?Is this a good research topic or we already have something in this direction

edited Feb 17 '15 at 16:19

user26857

asked Feb 17 '15 at 15:09

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Look up Sylow's theorems. – Mnifldz Feb 17 '15 at 15:12
Perhaps you meant to ask about subgroups of order $pn$? And assume that $\gcd(p,n)=1$? – hardmath Feb 17 '15 at 15:28
if $gcd(p,n)=1$? – pink floyd Feb 17 '15 at 15:31
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For abelian groups this has been discussed previously. It is not generally true, even if we assume $n$ is a prime distinct from $p$. E.g. Why $A_5$ has no subgroup of order $15$? – hardmath Feb 17 '15 at 16:09
Do you know the subgroups of $A_4$? – j.p. Feb 17 '15 at 17:36

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