Algebra: Non-isomorphic groups

[nuuskur]

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 G =\{0, g\}.
Then also g+g = 0 must be true, means g is its own opposite. (of order 2).

Now, how should I construct a group under some operation $*$: G' = \{e, g'\}, where $e$ is the unit/zero element (depending on operation) such that G and G' are not isomorphic?

G\cong G' iff there exists a bijective group homomorphism f: G\to G'
I can define f such that:
f(0) = e (satisfies one of the group homormorphism requirement)
f(g) = g'
Is bijection and

f(0+g) = f(g) = g' = e*g'
f(g+g) = f(0) = e = g'*g'
so G\cong G'

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?

 

[Samy_A]

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

Do you think that is possible?

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

I think that is not easy at all.

 

[pasmith]

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 G =\{0, g\}.
Then also g+g = 0 must be true, means g is its own opposite. (of order 2).

Now, how should I construct a group under some operation $*$: G' = \{e, g'\}, where $e$ is the unit/zero element (depending on operation) such that G and G' are not isomorphic?

G\cong G' iff there exists a bijective group homomorphism f: G\to G'
I can define f such that:
f(0) = e (satisfies one of the group homormorphism requirement)
f(g) = g'
Is bijection and

f(0+g) = f(g) = g' = e*g'
f(g+g) = f(0) = e = g'*g'
so G\cong G'

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

You can't. You are asked "how many non-isomorphic groups of two elements are there?" and you've shown that the answer is "all groups of order two are ismorphic to the cyclic group of order two."

 

[nuuskur]

EDIT: nope