Solve Group Theory Problem: (Z_4 x Z_4 x Z_8)/<(1,2,4)>

In summary: The Attempt at a SolutionI am not sure I have seen that proof...Maybe I need to use the Fundamental Homomorphism Theorem...to use this I need to find a group G' and a homomorphism phi such that ker(phi)=<(1,2,4)>. How would I figure out what G' is and what phi is though...anyone?anyone?
  • #1
ehrenfest
2,020
1
[SOLVED] group theory problem

Homework Statement


Classify the factor group (Z_4 cross Z_4 cross Z_8)/<(1,2,4)> according to the fundamental theorem of finitely generated abelian groups.


Homework Equations





The Attempt at a Solution


<(1,2,4)> has order 4 so the factor group has order 32, so there are seven possibilities:

(Z_2)^5
Z_32
(Z_2)^3 cross Z_4
Z_16 cross Z_2
Z_8 cross Z_2 cross Z_2
Z_2 cross Z_4 cross Z_4
Z_8 cross Z_4

Anyone have any ideas about how to do this without doing a lot of tedious calculations?
 
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  • #2
It seems evident to me: your relation says:

(1, 0, 0) + (0, 2, 0) + (0, 0, 4) = (0, 0, 0)

*shrug*


Maybe rewriting it as a quotient of Z³ would help?
 
  • #3
What is "it"?
 
  • #4
Let a,b,c be the generators of Z/4, Z/4, and Z/8, respectively. Then modding out by (1,2,4) is the same as saying that a=-2b-4c. So this means that a is completely determined by b and c, and hence doesn't matter anymore. Now the question is whether b and c can be whatever they want (between 0 and 3 and 0 and 7, respectively) and give different elements. I'll leave the rest to you.
 
  • #5
masnevets said:
Let a,b,c be the generators of Z/4, Z/4, and Z/8, respectively. Then modding out by (1,2,4) is the same as saying that a=-2b-4c. So this means that a is completely determined by b and c, and hence doesn't matter anymore.

I am kind of confused about this. Does Z/4 mean the same thing as Z_4 i.e. the integers mod 4? If so, then a=b=c=1, and 1 is not equal to -6.
 
  • #6
Can someone please elaborate on what masnevets is saying?
 
  • #7
anyone? this problem is killing me!
 
  • #8
please?
 
  • #9
Yes, Z/4 is the integers modulo 4. Yes, 1 is not equal to -6, but you're confusing what I mean now. a, b, and c mean (1,0,0), (0,1,0), and (0,0,1), respectively.
 
  • #10
So, you're just saying that the class of (1,0,0) is in the same as the class of (0,-2,-4) ? I agree. Those two elements are clearly in the same coset. But how does that help you classify the quotient group according to the fundamental theorem of finitely generated abelian groups?!
 
  • #11
The natural homomorphism from the group to the qotient is going to be onto, so the image of a set of generators is a set of generators.
 
  • #12
NateTG said:
The natural homomorphism from the group to the qotient is going to be onto, so the image of a set of generators is a set of generators.

I am not sure I have seen that proof...

But you're saying that the (1,0,0)+<(1,2,4)>,(0,1,0)+<(1,2,4)>,(0,0,1)+<(1,2,4)> will generate the quotient group?

Can someone just give me a concrete instruction so that I can make some progress classifying this group!

Maybe I need to use the Fundamental Homomorphism Theorem...to use this I need to find a group G' and a homomorphism phi such that ker(phi)=<(1,2,4)>. How would I figure out what G' is and what phi is though...
 
Last edited:
  • #13
anyone?
 
  • #14
anyone?
 

What is the group theory problem (Z_4 x Z_4 x Z_8)/<(1,2,4)>?

The group theory problem (Z_4 x Z_4 x Z_8)/<(1,2,4)> is a mathematical problem that involves finding the quotient group of the direct product of three groups, namely Z_4, Z_4, and Z_8, by the subgroup generated by the elements (1,2,4).

What is the purpose of solving this group theory problem?

The purpose of solving this group theory problem is to understand the structure and properties of the quotient group, which can provide insights into the original direct product group and its subgroups. This problem also has applications in various fields such as algebra, geometry, and physics.

What is the significance of the subgroup (1,2,4) in this problem?

The subgroup (1,2,4) is significant because it is the generator of the quotient group. This means that it plays a crucial role in defining the structure and elements of the quotient group. Its elements, when applied to the direct product group, generate the equivalence classes of the quotient group.

How can the group theory problem (Z_4 x Z_4 x Z_8)/<(1,2,4)> be solved?

The group theory problem (Z_4 x Z_4 x Z_8)/<(1,2,4)> can be solved by first finding the elements of the quotient group and then determining the group operation. This can be achieved by using the concept of cosets and the properties of the direct product and subgroup. Alternatively, computer programs and algorithms can also be used to solve this problem.

What are the possible applications of solving this group theory problem?

Solving the group theory problem (Z_4 x Z_4 x Z_8)/<(1,2,4)> has various applications in different fields. In algebra, it can be used to understand the structure of finite groups and their subgroups. In geometry, it can help in the classification of geometric figures and in physics, it has applications in the study of crystal structures and symmetry operations.

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