The number of groups of order n(upto isomorphism)is

Question

The number of groups of order $n$ (upto isomorphism)is

a)finite for all values of $n$

b)finite only for finitely many values of $n$

c)finite for infinitely many values of $n$

d)infinite for some values of $n$

I think option (b) is correct.please help me

See whether this helps you: http://math.stackexchange.com/questions/466395/number-of-distinct-groups-of-order-n-upto-isomorphism-for-a-fixed-integer-n — , Jan 06 '17 at 12:03
And hopefully you can choose more than one option because (a) implies (c) [and, less importantly, (b) implies (d)] :-) — Alex Macedo, Jan 06 '17 at 12:18
This question is similar to: Show that the number of nonisomorphic finite groups of order $n$ is at most $n^{n^2}.$. If you believe it’s different, please [edit] the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. — Robin, May 10 '25 at 21:17

score 11 · Answer 1 · answered Jan 06 '17 at 11:56

11

Note that on a set $G$ with $n$ elements, there are $n^{n^2}$ maps $G \times G \to G$, hence there are at most $n^{n^2}$ non-isomorphic groups of order $n$.

answered Jan 06 '17 at 11:56

martini

86,011

The number of groups of order n(upto isomorphism)is

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