## Arbitrarily large groups with no non-trivial subgroups?

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kcaze
Posts: 57
Joined: Sat Jul 24, 2010 2:23 am UTC

### Arbitrarily large groups with no non-trivial subgroups?

Is it possible to construct arbitrarily large groups with no subgroups that are non-trivial (the unit element and the entire group itself)? I've begun to learn some abstract algebra by following Pinter's A Book of Abstract Algebra but I'm not very far in and this problem occurred to me. So for now, I'm mostly interested in a yes or no answer because I think any proof will probably be over my head.

By the way, when does abstract algebra actually begin to get interesting? Right now, I know that there are different types of algebraic structures, but there doesn't actually seem to be much you can really do with them besides classifying sets as groups or not.

mr-mitch
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### Re: Arbitrarily large groups with no non-trivial subgroups?

Well, the smallest non-trival subgroup of a group G is the generating set of some element in G (by closure).

If it has no non-trivial subgroups, then every generating set must be G, yes? So the order of every element is |G|.

If the order of an element g is n, how do you work the order of an element g^k, k<=|G|? If all the orders are |G|, is there any relation between k and |G|?
What numbers have this property?

kcaze
Posts: 57
Joined: Sat Jul 24, 2010 2:23 am UTC

### Re: Arbitrarily large groups with no non-trivial subgroups?

Oh, I think I get it (after looking up what the order of an element meant). The order of g^k is only equal to |G| when k and |G| are relatively prime right? So as long as |G| is a prime number, we can find a group with no non-trivial subgroups right? More specifically, the group generated by a single element a so that the group is {a, a^2, a^3, ..., e}?

Is that right?

Token
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### Re: Arbitrarily large groups with no non-trivial subgroups?

The only groups with no non-trivial proper subgroups are the cyclic groups of prime order.
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skeptical scientist
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### Re: Arbitrarily large groups with no non-trivial subgroups?

Token wrote:The only groups with no non-trivial proper subgroups are the cyclic groups of prime order.

This is quite easy to prove. Take an arbitrary non-identity element of the group. Either it generates a non-trivial proper subgroup, or the whole group. If it generates the whole group, then the group is cyclic, and cyclic groups of non-prime order have proper subgroups.
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