Questioning Factor Groups: Understanding Properties and Theorems

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In summary, the conversation discusses various proofs and examples related to factor groups, including the fact that a normal subgroup is necessary for a factor group to have two elements, the structure of the alternating group A_4 and its subgroups, and the isomorphism of factor groups using Lagrange's theorem and the fundamental theorem of finitely generated abelian groups. It also touches upon the concept of finite groups being finitely generated.
  • #1
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I have a few question concerning factor groups.

1. In a proof for the fact that if a finite factor group G/N has 2 elements, then N is a normal subgroup of G, it says:
"For each a in G but not in H, both the left coset aH, and right coset Ha, must consist of all elements in G that are not in H".

Why is this so?

2. For the alternating group A_4, let H be a subgroup of order 6 (the point is to show there's no subgroup of order 6), it says that A_4/H = {H, sH} for some s in A_4 but not in H. Also, (H)(H) = H, (sH)(sH) = H. It then says, a in H implies a^2 in H, and b in sH implies b^2 in H. That is, the square of any element in A_4 must be in H.

Where does the last sentence come from? The line before that says the square of anything in H or sH is in H, so why does it then conclude this is also true for any element of A_4?

3. In an example, it says that (Z_4 x Z_6)/<(2,3)> has order 12 (just use lagrange's theorem), it then concludes that this factor group must be isomorphic to Z_4 x Z_3 or Z_2 x Z_2 x Z_3. (Z_n is the group {0, 1, ..., n-1}.)

I think it used the fundamental theorem of finitely generated abelian groups here. But that would require (Z_4 x Z_6)/<(2,3)> to be finitely generated. Is there any obvious way to see this? (It is obviously abelian).

We can see that (Z_4 x Z_6) = <(1,0) , (0,1)> so is finitely generated, but is there a theorem that says if G is finitely generated then so is its factor group G/N? (My textbook has a theorem that if G is cyclic then G/N is cyclic, but (Z_4 x Z_6) is not cyclic.) Or is every finite group finitely generated?
 
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  • #2
I took algebra last semester, and it is already fuzzy in my mind. See if the following helps.
  1. Two left cosets {H, aH}. Two right cosets {H, Ha}. G = H ∪ aH = H ∪ Ha. Hence, aH = Ha = G-H.
  2. |A4| = 12. |H| = 6. Do you see how the conclusion now follows from #1?
  3. If G is finite, then it is trivially finitely generated by all of itself.
 
  • #3
Tedjn said:
I took algebra last semester, and it is already fuzzy in my mind. See if the following helps.

[*]Two left cosets {H, aH}. Two right cosets {H, Ha}. G = H ∪ aH = H ∪ Ha. Hence, aH = Ha = G-H.
Ahh right. I forgot that the set of coset partition the group. Then the result is obvious.
Tedjn said:
[*]|A4| = 12. |H| = 6. Do you see how the conclusion now follows from #1?
I knew that #1 implied everything up until the last sentence. I'm still not sure where the last sentence comes from.
Tedjn said:
[*]If G is finite, then it is trivially finitely generated by all of itself.
[/LIST]
I see.
 
  • #4
Because A4 = H ∪ sH.
 
  • #5
Hey. Thanks for the help!
 
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FAQ: Questioning Factor Groups: Understanding Properties and Theorems

1. What is a factor group?

A factor group, also known as a quotient group, is a mathematical concept in group theory. It is a group that is formed by taking a subgroup of a larger group and considering all the elements in that subgroup as a single element in the factor group.

2. How is a factor group different from a normal subgroup?

A normal subgroup is a subgroup that is invariant under conjugation by all elements in the larger group. In contrast, a factor group is formed by taking a subgroup and "modding out" by the normal subgroup, essentially collapsing all elements in the normal subgroup into a single element in the factor group.

3. Can the order of a factor group be greater than the order of the original group?

Yes, the order of a factor group can be greater than the order of the original group. This can happen when the normal subgroup being factored out has a larger order than the subgroup it is being factored out of. In this case, the factor group will have more elements than the original group.

4. What is the significance of factor groups in mathematics?

Factor groups have many applications in mathematics, particularly in algebra and number theory. They are used to study the structure and properties of groups, and are essential in understanding topics such as symmetry, geometry, and abstract algebra.

5. How can factor groups be visualized?

Factor groups can be visualized as a "folding" or "wrapping" of a larger group onto a smaller subgroup. This can be represented as a diagram or a graph, with the elements of the factor group grouped together and connected to the corresponding elements in the larger group that they are being factored out of.

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