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## Homework Statement

How many non-isomorphic groups of two elements are there?

## Homework Equations

## The Attempt at a Solution

I don't understand exactly what we are being asked.

If we have a group of two elements under, say, addition, then [itex]G =\{0, g\}[/itex].

Then also [itex]g+g = 0[/itex] must be true, means [itex]g[/itex] is its own opposite. (of order 2).

Now, how should I construct a group under some operation ##*##: [itex]G' = \{e, g'\}[/itex], where ##e## is the unit/zero element (depending on operation) such that [itex]G[/itex] and [itex]G'[/itex] are

**not**isomorphic?

[itex]G\cong G'[/itex] iff there exists a bijective group homomorphism [itex]f: G\to G'[/itex]

I can define [itex]f[/itex] such that:

[itex]f(0) = e[/itex] (satisfies one of the group homormorphism requirement)

[itex]f(g) = g'[/itex]

Is bijection and

[itex]f(0+g) = f(g) = g' = e*g'[/itex]

[itex]f(g+g) = f(0) = e = g'*g'[/itex]

so [itex]G\cong G'[/itex]

What must I do to generate non-isomorphic groups of two elements?

In general, for any ##n##, how can I determine the number of non-isomorphic groups of ##n## elements?

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