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If a group has two generators but has a long composition series, can we conclude that something similar to wreath product is used somewhere? what other constructions can be used?

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I don't understand, apprently, the question: why do you think there's some relation between two generators, length of compositions series and wreath products? Try to give us more background, where does this come from, insights.... – DonAntonio Dec 20 '12 at 12:46
I am working with cayley graphs. If we have two generators (a,b), then the generators of the groups in the composition series are words of a and b. I am trying to find a way to generate these words. If the group is a direct product of other groups, then the number of generators increase. Wreath product gave a large group without much increase in the number of generators. – Ramyaa Dec 20 '12 at 12:58
The generators of any subgroup of any group in any number of generators are always words in the generators' letters, so what is special in two generators within this context? And again: what have wreath products to do here? – DonAntonio Dec 20 '12 at 13:05
I am trying to enumerate the words that form the generators of the groups in the composition series. When I try this with large groups with two generators, the enumeration of the words follow a pattern similar to that we get when wreath products are used. Since wreath products are one way to get a long composition series with small number of generators, I was wondering if wreath products are always needed. – Ramyaa Dec 20 '12 at 13:17

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