How can you tell whether 2 groups are isomorphic?

Suppose you had the groups

G

and

H

. How would you tell whether they were isomorphic? What about whether they were homomorphic?

I've seen definitions of homomorphic, such as

G

and

H

are homomorphic iff there exists a function

f:G \rightarrow H

, such that

f(ab)=f(a)f(b), \forall a,b \in G

, but I don't see how you would use this definition to tell whether 2 given groups were homomorphic. Any advice?

Thanks

(edited 8 years ago)

Reply 1

atsruser

Original post by Brian Moser

Suppose you had the groups

G

and

H

. How would you tell whether they were isomorphic? What about whether they were homomorphic?

I've seen definitions of homomorphic, such as

G

and

H

are homomorphic iff there exists a function

f:G \rightarrow H

, such that

f(ab)=f(a)f(b), \forall a,b \in G

, but I don't see how you would use this definition to tell whether 2 given groups were homomorphic. Any advice?

Thanks

You need to produce a function that is an isomorphism or homomorphism, but that essentially requires you to guess an appropriate function that looks suitable, and then show that it does indeed satisfy the definition of an iso/homomorphism.

Reply 2

Brian Moser

Original post by atsruser

And how are you supposed to know whether or not that function exists? You could be looking for one forever if there actually aren't any?

Reply 3

metaltron

Original post by Brian Moser

Suppose you had the groups

G

and

H

. How would you tell whether they were isomorphic? What about whether they were homomorphic?

I've seen definitions of homomorphic, such as

G

and

H

are homomorphic iff there exists a function

f:G \rightarrow H

, such that

f(ab)=f(a)f(b), \forall a,b \in G

, but I don't see how you would use this definition to tell whether 2 given groups were homomorphic. Any advice?

Thanks

It can be quite difficult to show two groups are isomorphic by finding an explicit bijective homomorphism, but if you wanted to check if two groups of order 4 (for example) are isomorphic then you can check if there is an element of order 4 (and so both are isomorphic to C4) or else it is isomorphic to C2xC2.

I don't know what you mean by your second question. If you wanted to see if there is a surjective homomorphism between G and H, you'd have to find one explicitly or find a quotient group G/K (for some subgroup K of G) isomorphic to H. Note the order of H has to divide the order of G (why?) .

(edited 8 years ago)

Reply 4

DFranklin

Original post by Brian Moser

I've seen definitions of homomorphic, such as

G

and

H

are homomorphic iff there exists a function

f:G \rightarrow H

, such that

f(ab)=f(a)f(b), \forall a,b \in G

, but I don't see how you would use this definition to tell whether 2 given groups were homomorphic. Any advice?I don't think this definition makes a lot of sense, since you can always define a homomorphism from G to H (just map

f(a) \to 1_H

for all a).

Reply 5

atsruser

Original post by Brian Moser

And how are you supposed to know whether or not that function exists? You could be looking for one forever if there actually aren't any?

Yes, that is true. But you may be able to see that two groups are not isomorphic, so you don't even need to start. In general though, the "group isomorphism problem" is undecidable, so there is no algorithm that will always terminate when presented with the problem of showing that two groups are isomorphic:

https://en.wikipedia.org/wiki/Group_isomorphism_problem

You can show that two groups A and B are not isomorphic by finding a property that is present in group A, that is preserved by isomorphism (e.g. order, abelianness, ..), and which is not present in group B.

For finite groups though, in principle, I guess you can just crank out every mapping between the two, and check them until you find an isomorphism or not.

Reply 6

Brian Moser

Original post by atsruser

Okay, thanks. To put that into practice though, say you had the group

\mathbb{Z}_{6}

and the group

G

of rotations preserving a regular tetrahedron. Are these isometric, and why?

Thanks again. Really appreciate the advice from everyone.

(edited 8 years ago)

Reply 7

Gregorius

Original post by atsruser

For finite groups though, in principle, I guess you can just crank out every mapping between the two, and check them until you find an isomorphism or not.

Even limiting the question to one concerning finite groups, one is left with a fascinating and - as far as I can find out - open problem in complexity theory.

See this post on Mathoverflow and this blog post that summarizes Tarjan's algorithm.

Reply 8

Gregorius

Original post by Brian Moser

Okay, thanks. To put that into practice though, say you had the group

\mathbb{Z}_{6}

and the group

G

of rotations preserving a regular hexagon. Are these isometric, and why?

After eliminating the obvious (e.g. group order), find a set of generators of G, attempt to match those generators with elements of H, and then see if the matching induces an isomorphism. (Tarjan's algorithm).