Find Center of Groups of Order 8

In summary, to find the center of groups of order 8, one can use the class equation and determine that the only possible orders for the center are 1, 2, or 8. However, since the elements in the center are already accounted for in the first term, the indices of the remaining elements must be even. Therefore, it is impossible for the center to have order 1, leading to the conclusion that the center of a group of order 8 must have order 2 or 8.
  • #1
Fessenden
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How to find the center of groups of order 8?
 
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  • #2
Fessenden said:
How to find the center of groups of order 8?

HINT: Start by proving that if $G/Z(G)$ is cyclic then $G=Z(G)$.

This means that if $|G|=8$ then $|Z(G)|=1$, $2$ or $8$. Why can't it be $1$?
 
  • #3
another way to put this is: if G is abelian, then Z(G) = G (which isn't very interesting). i believe the hint Swlabr is trying to steer you to is the class equation:

$|G| = |Z(G)| + \sum_{a \not\in Z(G)} [G:N(a)]$ where N(a) is the normalizer (or, in some texts, the centralizer) of the element a.

note each index [G:N(a)] must divide 8, and since we have the elements of Z(G) in the first term, and the a's are NOT in the center, each [G:N(a)] has to be either 2, or 4 (the only conjugacy classes of size 1 are elements of the center). in any case, each [G:N(a)] is an even number. is there an integer solution to:

8 = 1 + 2k?
 

FAQ: Find Center of Groups of Order 8

What is the "Find Center of Groups of Order 8" problem?

The "Find Center of Groups of Order 8" problem is a mathematical problem that involves finding the center of a group with 8 elements. The center of a group is the set of elements that commute with every element in the group, meaning that their order does not matter when performing operations on the group.

Why is finding the center of groups of order 8 important?

Finding the center of groups of order 8 is important in the study of abstract algebra and group theory. The properties and structure of the center can provide insights into the behavior of the group and help solve other related problems.

How do you find the center of groups of order 8?

To find the center of a group with order 8, you need to first list out all the elements of the group and then perform the operation of composition on each pair of elements. The elements that commute with every other element will form the center of the group.

What are some properties of the center of groups of order 8?

The center of a group with order 8 has several properties, including being a subgroup of the original group, containing the identity element, and being abelian (meaning the order of the elements does not affect the outcome of operations). The center may also have a non-trivial size, meaning it is not just the identity element.

Are there any practical applications of the "Find Center of Groups of Order 8" problem?

While the "Find Center of Groups of Order 8" problem may seem abstract, it has real-world applications in cryptography and coding theory. The properties of the center can be used to create secure encryption algorithms and error-correcting codes.

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