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.

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

**Moderators:** gmalivuk, Moderators General, Prelates

### 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?

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?

### 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?

Is that right?

### 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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### 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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