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lmn123
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Homework Statement
For each m >= 2, find a group with a composition series of length 1 with a subgroup of length m.
Homework Equations
Simple groups iff length 1.
If G is abelian of order p1^k1...pr^kr, then length G = k1 + ... + kr
If G has a composition series and K is normal subgroup of G, then Length G = Length K + Length G/K
The Attempt at a Solution
I don't really see how this is even possible considering the fact that simple groups have length 1. So this means that G must be a simple group. I do not understand how to get a group of length m from a simple group.
I think I'm getting somewhere with this:
What I was thinking was have G = the cyclic group of order p. Then G would have length 1
Then I thought to Let H be a subgroup of G such that |G|=p^m, so if G=<a>, st |a|=p, then H=<a^p^(m+1)>
But this doesn't really work because the subgroup would have higher order than G right? Am I missing something, like modular arithmetic?
Any light you guys could shed would be appreciated
Thanks