Number of Isomorphisms f from G to G' of Order 8 Cyclic Groups

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In summary, there are 32 possible isomorphisms from a cyclic group of order 8 to another cyclic group of order 8, based on the number of elements of each order and the possible choices for the isomorphism f.
  • #1
sutupidmath
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The question is this: How many isomorphisms f are there from G to G' if G and G' are cyclic groups of order 8?

My thoughts:

Since f is an isomorphism, we know that it prserves the identity, so f:e-->e', e identity in G, e' identity in G'.

Also f preserves the order of each element. That is if o(a)=k=>o(f(a))=k

SO, i thought that f will send the el of the same order in G to the corresponding elements of the same order in G'.

Let G=[a], and G'=. so it means that there are 4 el in G that have order 8 ( the generators of G, a, a^3, a^5,a^7), so there are 4 possibilities for these elements, hense by keeping the other el. fixed we would have 4^4 isomorphisms.

But also we have 2 el of order 4, (a^2, and a^6) so there are two possibilities for these elements to be mapped into G' by f, so if the other el are fixed we would have 2^2 mappings.

Does this mean that the total nr of such isomorphisms is 4^4+2^2? Or am i totally on the wrong way?

Any suggestions would be appreciated.
 
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  • #2
f must satisfy f(a)=bk, some 1 <= k <= 7, and f is completely determined by the choice of k.
You should be able to see how many possible choices there are for k.
 

1. What is an isomorphism in the context of cyclic groups?

An isomorphism is a bijective function between two groups that preserves the group structure. In the context of cyclic groups, it means that the isomorphism maps elements from one cyclic group to another in a way that maintains the order and operations of the groups.

2. How do you determine the number of isomorphisms between two cyclic groups of order 8?

The number of isomorphisms between two cyclic groups of order 8 is equal to the number of generators of the first group, which is also the number of elements with order 8 in the second group. In the case of cyclic groups of order 8, there are 4 elements with order 8, so there are 4 possible isomorphisms.

3. Can there be more than one isomorphism between two cyclic groups of order 8?

No, there can only be one isomorphism between two cyclic groups of order 8. This is because the number of isomorphisms is determined by the number of generators of the first group, which is unique for each cyclic group.

4. What is the significance of the number of isomorphisms between cyclic groups of order 8?

The number of isomorphisms between cyclic groups of order 8 is significant because it reflects the number of elements with order 8 in the second group. It also shows the relationship between the two groups and how they are structurally similar.

5. Are there any other factors that can affect the number of isomorphisms between cyclic groups of order 8?

Yes, the number of isomorphisms can also be affected by the choice of generators for the first group. Different choices of generators can result in different numbers of isomorphisms. However, as long as the generators have the same order, the number of isomorphisms will remain the same.

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