Math 2.0

example

The integers modulo 2 and the group of signs are isomorphic

Example

Let Z/2Z={0,1}\mathbb{Z}/2\mathbb{Z} = \{0, 1\} be the integers modulo 22 with addition. Let {1,−1}\{1, -1\} be a group under multiplication. Define f(0)=1f(0) = 1 and f(1)=−1f(1) = -1. Then ff is a bijection. We check the four products: f(0+0)=1=1⋅1f(0 + 0) = 1 = 1 \cdot 1, f(0+1)=−1=1⋅(−1)f(0 + 1) = -1 = 1 \cdot (-1), f(1+0)=−1=(−1)⋅1f(1 + 0) = -1 = (-1) \cdot 1 and f(1+1)=f(0)=1=(−1)⋅(−1)f(1 + 1) = f(0) = 1 = (-1) \cdot (-1). Thus ff is a homomorphism and a bijection, so ff is an isomorphism.

Remarks

Both groups are cyclic of order 22. This agrees with the result that cyclic groups of the same order are isomorphic.

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Sources

  1. Web https://en.wikipedia.org/wiki/Group_isomorphism, Cyclic groups

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