"nontrivial subgroups" (terminology question)

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skullturf
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"nontrivial subgroups" (terminology question)

Postby skullturf » Fri Feb 11, 2011 10:05 pm UTC

If I have a group G and I say "consider its nontrivial subgroups" with no further explanation, what do I mean?

What are the "trivial subgroups" I'm ignoring?
(a) the group containing only the identity
(b) the group containing only the identity, and G itself.

A rather cursory Google search seems to indicate some degree of consensus. Do you think my question has a generally agreed-upon answer? Or do you think it's ambiguous?

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z4lis
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Re: "nontrivial subgroups" (terminology question)

Postby z4lis » Fri Feb 11, 2011 10:09 pm UTC

If you made it a poll, I would vote (b). But I'm also an undergraduate who's only had a year of algebra, so I'm certainly not in touch with current usage in the literature.
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Harg
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Re: "nontrivial subgroups" (terminology question)

Postby Harg » Fri Feb 11, 2011 10:14 pm UTC

I'd say (a) is the usual meaning. You often see "proper notrivial subgroups" if you want to exclude (b).

theorigamist
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Re: "nontrivial subgroups" (terminology question)

Postby theorigamist » Fri Feb 11, 2011 10:23 pm UTC

The group consisting of a single element, {e}, is called the trivial group. Usually, nontrivial subgroup means subgroup not isomorphic to the trivial group (that is, (a)). Frequently, nontrivial subgroup is used to mean proper, nontrivial subgroup. You shouldn't worry too much about the distinction. If a problem just says "consider the nontrivial subgroups of G and prove X about them", then check if X is true of G itself. If not, then "nontrivial" means "proper and nontrivial" for the purposes of that problem.

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Eebster the Great
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Re: "nontrivial subgroups" (terminology question)

Postby Eebster the Great » Mon Feb 14, 2011 2:11 pm UTC

I often see "nontrivial" in the context of groups to be analogous to "nonempty" in the context of sets, so (a).

Hix
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Re: "nontrivial subgroups" (terminology question)

Postby Hix » Tue Feb 15, 2011 12:26 am UTC

Definitely (a)

The trivial subgroup of any group G is the subgroup with just one element. So a nontrivial subgroup is any other subgroup. If G itself is not supposed to be considered, you should specify "proper", as others have indicated.

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Re: "nontrivial subgroups" (terminology question)

Postby Token » Tue Feb 15, 2011 8:50 pm UTC

Hix wrote:The trivial subgroup of any group G is the subgroup with just one element.

You say that like maths terminology is standardised. I seem to recall (though I may easily be wrong) that one of my group theory lecturers for my master's explicitly enumerated the trivial subgroups of a group G as {e} and G itself. The problem lies in whether the word "trivial" is used to mean triviality as a group (in which case, only {e}) or triviality as a group-subgroup pair (and having them both be the same is certainly trivial). It's ambiguous - if you're reading, don't assume, and if you're writing, be explicit.
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skullturf
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Re: "nontrivial subgroups" (terminology question)

Postby skullturf » Tue Feb 15, 2011 9:27 pm UTC

My cursory searching seemed to indicate that it's almost standard to use "trivial subgroup" to refer only to the one-element group.

But I agree with Token: the words "trivial subgroup" are close to "trivially a subgroup", in which case it's not that far-fetched to interpret it as including the whole group. If you're writing about a situation where it matters, it's probably better to be explicit and not assume.


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