Group theory and generalizations Example

The integers modulo 2 and the group of signs are isomorphic

Builds on Integers modulo n Group isomorphism

Example

Let $\mathbb{Z}/2\mathbb{Z} = \{0, 1\}$ be the integers modulo $2$ with addition. Let $\{1, -1\}$ be a group under multiplication. Define $f(0) = 1$ and $f(1) = -1$ . Then $f$ is a bijection. We check the four products: $f(0 + 0) = 1 = 1 \cdot 1$ , $f(0 + 1) = -1 = 1 \cdot (-1)$ , $f(1 + 0) = -1 = (-1) \cdot 1$ and $f(1 + 1) = f(0) = 1 = (-1) \cdot (-1)$ . Thus $f$ is a homomorphism and a bijection, so $f$ is an isomorphism.

Remarks

Both groups are cyclic of order $2$ . This agrees with the result that cyclic groups of the same order are isomorphic.

Reading path

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Group concept
Integers modulo n concept
Group homomorphism definition
Group isomorphism definition
The integers modulo 2 and the group of signs are isomorphic example

Tag	`7FCN2KNWC0`
URI	`m2://7FCN2KNWC0`
Kind	example
Writing	W0 stub
MSC	20A05
Instance of	Group isomorphism
Illustrates	Cyclic groups of the same order are isomorphic
Uses	Integers modulo n, Group isomorphism

The integers modulo 2 and the group of signs are isomorphic

Example

Remarks

Reading path

Further reading

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