Is a group isomorphic to one of its realizations? Suppose I give you an "abstract group" G with the exact same multiplication table as the set {pm 1, pm i, pm j, pm k} (quaternions) under multiplication. And someone says find a homomorphism for this abstract group.

clasicaacyx

clasicaacyx

Answered question

2022-09-25

Is a group isomorphic to one of its realizations?
Suppose I give you an "abstract group" G with the exact same multiplication table as the set { ± 1 , ± i , ± j , ± k } (quaternions) under multiplication. And someone says find a homomorphism for this abstract group.
Since I know it follows the same multiplication table as the set I gave above can I suggest the appropriate bijection for those two groups, and say I've constructed the homomorphism?

Answer & Explanation

Micah Hobbs

Micah Hobbs

Beginner2022-09-26Added 8 answers

Explanation:
A group is well defined in mathematics and there is no significant distinction between an "abstract" group and a "concrete" group. A group is an ordered pair (G,⋅) where G is a set and ⋅ is a binary operation G × G G satisfying the appropriate axioms.

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