Group Theory: Isomorphism between Z_3 x Z_4 & Z_12

The confusion may arise because these generators are not as easily identifiable as the generators of Z_12, but they still exist. The statement about phi(n) is true in general and can be used to determine the number of generators for any cyclic group. In summary, Z_3 cross Z_4 is isomorphic to Z_12, meaning they have the same structure and properties, and both have four generators. The statement about the number of generators being equal to phi(n) is true in general.
  • #1
ehrenfest
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[SOLVED] group theory

Homework Statement


My book says that Z_3 cross Z_4 is isomorphic to Z_12, which I am confused about because
Z_3 cross Z_4 has four different generators and Z_12 only has 1.

EDIT: wait that is not true, Z_12 has generators 1,5,7,11
It is probably true in general that the number of generators of Z_n is equal to phi(n) where phi is the Euler phi function.

Homework Equations


The Attempt at a Solution

 
Last edited:
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  • #2
Since Z_3 cross Z_4 is isomorphic to Z_12, it means that they have the same properties and structure. This means that Z_3 cross Z_4 also has four generators, which are (1,0), (0,1), (2,0), (0,3).
 

1. What is Group Theory?

Group Theory is a branch of mathematics that studies the properties of groups, which are mathematical structures that consist of a set of elements and a binary operation that combines any two elements to produce a third element.

2. What is an isomorphism?

An isomorphism is a structure-preserving mapping between two mathematical structures. In the context of Group Theory, it refers to a mapping between two groups that preserves the group structure, meaning that the operation and properties of the groups are maintained.

3. What are Z3 x Z4 and Z12?

Z3 x Z4 and Z12 are two examples of finite groups. Z3 x Z4 is the direct product of the group of integers modulo 3 and the group of integers modulo 4. Z12 is the group of integers modulo 12.

4. How do you determine if there is an isomorphism between Z3 x Z4 and Z12?

In order for there to be an isomorphism between two groups, they must have the same order (number of elements). In this case, both Z3 x Z4 and Z12 have 12 elements, so it is possible for an isomorphism to exist. To determine if there is an isomorphism, we can examine the group tables and see if there is a one-to-one correspondence between the elements of the groups.

5. What is the isomorphism between Z3 x Z4 and Z12?

The isomorphism between Z3 x Z4 and Z12 is given by the mapping: (a,b) → 3a + 4b (mod 12). This mapping preserves the group structure and is a one-to-one correspondence between the elements of the groups, making Z3 x Z4 and Z12 isomorphic.

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