Direct product of two semi-direct products

Thread starter Cairo
Start date Mar 15, 2021
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In summary, the direct product of two semi-direct products is a mathematical operation that combines two semi-direct products into a single structure denoted by the symbol ⋆. It inherits the properties of both semi-direct products and is calculated by taking the Cartesian product of the underlying sets and defining the operation on the resulting elements. This operation is important in abstract algebra and group theory and has real-world applications in physics, chemistry, and computer science.

Mar 15, 2021

Cairo

Homework Statement: I need to find the number of elements and conjugacy classes for the direct product.

Relevant Equations: $$G=(C_7:C_3\ )\times(C_{13}:C_3\ )$$

After finding the number of elements for this group, how do I extend the argument to $$p,q\equiv1\left(mod\ 3\right)$$, where $$G=(C_p:C_3\ )\times(C_q:C_3\ )$$Any help appreciated.

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Mar 15, 2021

fresh_42

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What is ##C_p## and what is ##C_7:C_3##?

Mar 15, 2021

Cairo

$$C_7 : C_3$$ is a semi-direct product.

$$C_p$$ is as described. A prime, congruent to 1 mod 3.

Last edited: Mar 15, 2021

1. What is the direct product of two semi-direct products?

The direct product of two semi-direct products is a mathematical operation that combines two semi-direct products into a new group. It is denoted by the symbol × and is defined as the set of all possible combinations of elements from the two semi-direct products, with an operation that combines the elements in a specific way.

2. How is the direct product of two semi-direct products different from the direct product of two groups?

The direct product of two semi-direct products is different from the direct product of two groups because in a semi-direct product, the two groups are not necessarily normal subgroups of the direct product. This means that the operation used to combine the elements of the two semi-direct products is not necessarily commutative.

3. What is the significance of the direct product of two semi-direct products in mathematics?

The direct product of two semi-direct products is significant in mathematics because it allows for the creation of new groups with unique properties. It also helps in studying the structure of groups and understanding their relationships with other groups.

4. Can the direct product of two semi-direct products be commutative?

No, the direct product of two semi-direct products is not necessarily commutative. This is because the two semi-direct products may have different operations, and the operation used to combine them may not be commutative.

5. How is the direct product of two semi-direct products calculated?

The direct product of two semi-direct products is calculated by taking the Cartesian product of the two semi-direct products and then defining an operation on the resulting set. This operation must satisfy certain conditions, such as associativity and the existence of an identity element, in order for the direct product to be a valid group.