Arbitrarily large groups with no non-trivial subgroups?

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

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

Arbitrarily large groups with no non-trivial subgroups?

Postby kcaze » Wed Jun 08, 2011 1:51 pm UTC

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
Posts: 477
Joined: Sun Jul 05, 2009 6:56 pm UTC

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

Postby mr-mitch » Wed Jun 08, 2011 2:01 pm UTC

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?

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

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

Postby kcaze » Wed Jun 08, 2011 2:17 pm UTC

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
Posts: 1481
Joined: Fri Dec 01, 2006 5:07 pm UTC
Location: London

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

Postby Token » Wed Jun 08, 2011 2:31 pm UTC

The only groups with no non-trivial proper subgroups are the cyclic groups of prime order.
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.

User avatar
skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
Location: San Francisco

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

Postby skeptical scientist » Wed Jun 08, 2011 2:54 pm UTC

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.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson


Return to “Mathematics”

Who is online

Users browsing this forum: No registered users and 9 guests