Proving Isomorphism of Heisenberg Group over Finite Field

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In summary, the conversation discusses the subgroup H of GL(3, \mathbb{Z}_3) and the task of proving that Z(H) is isomorphic to \mathbb{Z}_3 and H/Z(H) is isomorphic to \mathbb{Z}_3 \times \mathbb{Z}_3. The approach mentioned involves defining a mapping from Z(H) to \mathbb{Z}_3 and calculating the product AC and CA and equating them to find the elements of Z(H). It is noted that these groups are also known as the Heisenberg group over a finite field.
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Homework Statement


Let H be the subgroup of [tex]GL(3, \mathbb{Z}_3)[/tex] consisting of all matrices of the form [tex]\left[ \begin{array}{ccc} 1 & 0 & 0 \\ a & 1 & 0 \\ b & c & 1 \end{array} \right][/tex], where [tex]a,b,c \in \mathbb{Z}_3[/tex]. I have to prove that Z(H) is isomorphic to [tex]\mathbb{Z}_3[/tex] and that [tex]H/Z(H)[/tex] is isomorphic to [tex]\mathbb{Z}_3 \times \mathbb{Z}_3[/tex].


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The Attempt at a Solution


I'm really not sure how to begin with this. I started by taking two arbitrary matrices h and k from H and doing hk = kh to see what a matrix in Z(H) would have to look like, but I didn't really get anywhere with that. My initial instinct would be to just define a mapping from Z(H) to [tex]\mathbb{Z}_3[/tex], but I'm not sure how to do that, since I can't figure out what's in Z(H). Is there a better way to do this?
 
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These groups are known in literature as Heisenberg group over a finite field. It is generally written in the upper triangular matrix form. But that does not make any difference.
So let me define a typical element of this field as [tex]A=\langle a,b,c\rangle[/tex] where a, b, c have the same meaning and order as you have written in the definition of the group. Let [tex]C=\langle x,y,z\rangle\in Z(H)[/tex]. Calculate [tex] AC[/tex] and [tex] CA [/tex], and equate them...


Edited by Hurkyl: please don't give complete solutions to problems...
 
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FAQ: Proving Isomorphism of Heisenberg Group over Finite Field

What is the definition of isomorphism in the context of groups?

Isomorphism is a mathematical concept that describes the relationship between two groups that have the same structure. In other words, two groups are isomorphic if there exists a one-to-one mapping between their elements that preserves the group operation and the identity element.

How can you prove that two groups are isomorphic?

To prove that two groups are isomorphic, you need to show that there exists a bijective function (also known as an isomorphism) between the two groups that preserves the group operation. This means that for every element in one group, there exists a corresponding element in the other group that has the same properties and vice versa.

What are some common properties that isomorphic groups share?

Isomorphic groups share many properties, including the number of elements, the order of the elements, the presence of a unique identity element, and the existence of inverses for each element. They also have the same group operation, which means that the outcome of the operation on two elements in one group will be the same as the outcome of the operation on their corresponding elements in the other group.

Can you prove that two groups are not isomorphic?

Yes, it is possible to prove that two groups are not isomorphic. One way to do this is by showing that one group has a property that the other group does not have. For example, if one group is cyclic and the other is not, they cannot be isomorphic. Another way is by showing that there is no bijective function between the two groups that preserves the group operation and identity element.

Why is proving groups isomorphic important in mathematics?

Proving groups isomorphic is important because it allows us to understand the underlying structure and relationships between different groups. It also helps us solve problems in different areas of mathematics, such as group theory, abstract algebra, and number theory. Additionally, isomorphism can be applied to other mathematical structures, such as rings and fields, to determine their similarities and differences.

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