# A set of non-isomorphic finite groups is a finite set

Let F= set of all non-isomorphic groups of order n where n>=2. I want to show that F is a finite set.

I want to use the fact: Every group |G|=n is an isomorphic to a subgroup of Sn. But i don't know how.

Can anyone give me a direction please? Thank you

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How many subgroups of $S_n$ are there? (Can there be infinitely many?) – Quinn Culver Dec 14 '12 at 16:18
How many distinct multiplication tables can you write down on a set of size $n$? – Alex B. Dec 14 '12 at 16:19

## 1 Answer

Show that a group is finite iff it has a finite number of subgroups.

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okay so i use the fact above to show that Sn has finite # of subgroups and all non-isomorphic groups in F are isomorphic to atleast one subgroup of Sn and therefore there are finitely many non-isomorphic groups in F, therefore F is a finite set! – d13 Dec 14 '12 at 16:34
Indeed so, just as Alex and Quinn already hinted in their comments. – DonAntonio Dec 14 '12 at 16:36
thanks a lot for ur help!! – d13 Dec 14 '12 at 16:37