Abstract algebraLesson 1 of 4 in the course Group homomorphisms and quotients

Group Homomorphisms: Classifications, Kernels, and Images

5 flashcards · 5 quiz questions · 5 match pairs · 3 fill-in sentences · English

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Summary

A group homomorphism is a function between groups that preserves their operation. This idea lets us compare groups by how their elements combine, even when the groups use different operations or have differently named elements.

Let (G,∗)(G, *) and (H,⋅)(H, \cdot) be groups. A function h:G→Hh:G\to H is a group homomorphism if, for every u,v∈Gu,v\in G,

The operation on the left is the operation in GG; the operation on the right is the operation in HH. The condition says that applying hh after combining two elements gives the same result as applying hh to each element and then combining their images.

All flashcards

What is a group homomorphism?
A function h:G→Hh:G\to H between groups that preserves the group operation: h(u∗v)=h(u)⋅h(v)h(u*v)=h(u)\cdot h(v) for every u,v∈Gu,v\in G.
What happens to the identity under a group homomorphism?
The identity of the domain maps to the identity of the codomain: h(eG)=eHh(e_G)=e_H.
How are inverses preserved?
The inverse of an element maps to the inverse of its image: h(u−1)=h(u)−1h(u^{-1})=h(u)^{-1}.
What distinguishes an isomorphism?
An isomorphism is a bijective group homomorphism. Its inverse is also a homomorphism, so it identifies the groups as structurally equivalent.
What are an endomorphism and an automorphism?
An endomorphism is a homomorphism from a group to itself. An automorphism is an endomorphism that is bijective.

Quiz questions

  1. A group homomorphism is injective but not necessarily surjective. What is it called?Answer: A monomorphism.
  2. A group homomorphism is surjective but not necessarily injective. What is it called?Answer: An epimorphism.
  3. If h:G→Hh:G\to H and k:H→Kk:H\to K are group homomorphisms, what can be said about k∘hk\circ h?Answer: It is a homomorphism from GG to $K.
  4. For a group homomorphism h:G→Hh:G\to H, what subgroup property does ker⁡(h)\ker(h) always have?Answer: It is a normal subgroup of GG.
  5. If GG and HH are abelian groups, what operation makes Hom⁡(G,H)\operatorname{Hom}(G,H) an abelian group?Answer: Pointwise addition of homomorphisms.

Based on the Wikipedia article Group homomorphism, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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