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I'd like to find easily-accessible composition series of an interesting lengths say 10 or so (to study some theorems about subgroup series experimentally). I'm thinking the integer mod unit groups, that is

##\mathbb{Z}_n^*=Z_0\rhd Z_1\rhd Z_2\rhd \cdots \rhd Z_9\rhd Z_{10}=\{1\}##. Would this be a relatively simple set of groups to do so with?

That's seems an interesting question: What is the smallest ##n## such that ##\mathbb{Z}_n^*## has a composition series of length ##10##? Or for that matter, what is the set of minimum such sizes for a range of ##n##? I have no idea and to make matters worst, I do now know how to easily find the set ##\{Z_i\}## other than number-crunching or I guess figure out maybe how to do it in GAP.

Is there a standard way to find these series? May be a GAP command to do so. Don't know yet -- not an easy program to use I think.

Thanks for reading,

Jack

Edit:

Ok just thought of something: A normal subgroup ##K\lhd G## is a maximal normal subgroup iff ##G/K## is simple. Ok then, they're all normal in ##Z_n^*## so then I guess just find the largest one in ##Z_n^*##, then find the largest one in that one, largest one in that one and so on until I get to 1. Not sure GAP can handle this though for large groups.

. . . I'm thinking in the millions to get it to 10.

Here's a start:

I interpret that to mean: ##S_{10}\rhd A_{10}\rhd \{1\}## and I think in fact we can write ##S_n\rhd A_n\rhd \{1\}##. Can't figure out how to code integer mod groups though.Code (Text):

gap> DisplayCompositionSeries(Group((1,2,3,4,5,6,7,8,9,10),(1,2)));

G (2 gens, size 3628800)

| Z(2)

S (8 gens, size 1814400)

| A(10)

1 (0 gens, size 1)

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# How to find composition series of Z_n^* of length k?

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