Proving the nonisomorphic groups of order 16 are indeed, not isomorphic.

[Armoneus]

Homework Statement

Find at least 7 pairwise non-isomorphic groups of order 16, and prove that no two among them are isomorphic.

Homework Equations

I found 7 nonisomorphic groups, but I just am having trouble how to precisely prove they are not isomorphic...

The Attempt at a Solution

(Z16), (Z8 x Z2), (Z4 x Z4), (Z4 x Z2 x Z2), (Z2 x Z2 x Z2 x Z2)
(D8), (D4 x Z2)

but how to prove non-isomorphic character...?

 

[mighty2000]

Hi there!

I can offer you some guidance on how to prove that these groups are non-isomorphic. First, let's define what it means for two groups to be isomorphic. Two groups G and H are isomorphic if there exists a bijective function f: G -> H such that for all x,y in G, f(xy) = f(x)f(y). In other words, an isomorphism is a function that preserves the group structure.

To prove that two groups are non-isomorphic, we need to show that there does not exist such a bijective function between them. One way to do this is by showing that the groups have different properties or structures that cannot be preserved by an isomorphism.

For example, let's take the groups (Z16) and (Z8 x Z2). (Z16) is the cyclic group of order 16, while (Z8 x Z2) is the direct product of the cyclic groups of order 8 and 2. We can show that these groups are not isomorphic by looking at their element orders. In (Z16), the only elements of order 2 are 8 and 16, while in (Z8 x Z2), there are 8 elements of order 2. This difference in element orders means that there cannot exist a bijective function between the two groups, and therefore they are not isomorphic.

Similarly, you can compare the other groups you have found by looking at their element orders, subgroups, or other properties. This will help you prove that they are not isomorphic.

I hope this helps! Let me know if you have any further questions.