# Find all nonisomorphic abelian gps

• mclove
In summary, an abelian group is a mathematical structure with a commutative binary operation. Two abelian groups are isomorphic if there exists a bijective map that preserves the group structure, while they are nonisomorphic if there is no such map. To find all nonisomorphic abelian groups, one lists all possible combinations and eliminates isomorphic groups. This is important for understanding the different structures and applications of abelian groups.

## Homework Statement

- List all nonisomorphic abelian groups of order 2^3 3^2 5

## The Attempt at a Solution

- Z_2^3 * Z_3^2 * Z_5 = (iso) Z_360.
Z_2^3 * Z_3 * Z_3 * Z_5 = (iso) Z_3 * Z_100
.
.
.
I know ,
but why Z_360 , Z_3 * Z_100 are nonisomorphic to 2^3 3^2 5 ?

Because Z_360 == Z_2^3 * Z_3^2 * Z_5, but Z_3 * Z_100 == Z_2^3 * Z_3 * Z_3 * Z_5.

Sorry, I don's understand it..
Z_2^3 * Z_3 * Z_3* Z_5 == Z_8 * Z_9 * Z_5 Is this nonabelian groups of order
2^33^25 ??

## 1. What is an abelian group?

An abelian group is a mathematical structure consisting of a set of elements and a binary operation (usually denoted by +) that is commutative, meaning the order in which the elements are combined does not affect the result. In other words, for any two elements a and b in the group, a + b = b + a.

## 2. What does it mean for two abelian groups to be isomorphic?

Two abelian groups are isomorphic if there exists a bijective map between them that preserves the group structure, meaning the operation and identity elements are preserved. In other words, if there is a way to rename the elements of one group to match the elements of the other group in a way that maintains the group's operation and identity elements, then the two groups are isomorphic.

## 3. What does it mean for two abelian groups to be nonisomorphic?

Two abelian groups are nonisomorphic if there is no way to rename the elements of one group to match the elements of the other group in a way that maintains the group's operation and identity elements. In other words, the two groups are structurally different and cannot be mapped onto each other while preserving the group structure.

## 4. How do you find all nonisomorphic abelian groups?

To find all nonisomorphic abelian groups, one must first list all possible combinations of elements and operations that satisfy the abelian group properties. Then, using the concept of isomorphism, groups with the same structure or those that can be mapped onto each other while preserving the group structure are considered isomorphic and can be eliminated from the list. The remaining groups are then the nonisomorphic abelian groups.

## 5. Why is it important to find all nonisomorphic abelian groups?

Finding all nonisomorphic abelian groups is important because it helps us understand the different possible structures and combinations of elements and operations that satisfy the abelian group properties. This knowledge can then be applied in various areas of mathematics, such as group theory, number theory, and algebraic geometry. Additionally, it allows us to classify and organize abelian groups, making it easier to study their properties and relationships with other mathematical structures.

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