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