Homomorphism
A map φ:G→H satisfying φ(ab)=φ(a)φ(b). It preserves identity and inverses.
Find a concept
REFERENCE SHEET
Homomorphisms & quotients
A quotient identifies precisely the information a homomorphism forgets.
Before this Groups & subgroups, Sets & maps
G / ker φ ≅ im φ
Definitions
Kernel
ker φ={g:φ(g)=e}. It is a normal subgroup of G.
Normal subgroup
N◁G means gNg⁻¹=N for all g. Then coset multiplication (gN)(hN)=ghN is well-defined.
Key theorems
First isomorphism theorem
For any group homomorphism φ:G→H, the map g ker φ ↦ φ(g) is a well-defined group isomorphism G/ker φ→im φ.
Injectivity criterion
A homomorphism is injective exactly when its kernel is {e}.
Notable applications
Parity of permutations
The sign homomorphism Sₙ→{±1} has kernel Aₙ; for n≥2 the quotient Sₙ/Aₙ is a group of order 2.
Linear algebra bridge
The linear-map analogue identifies V/ker T with im T.
Work it out
Try before revealing01 / Compute
For φ:ℤ→ℤ/5ℤ given by reduction mod 5, find the kernel and quotient.
Hint
Which integers map to zero?
Solution
ker φ=5ℤ and im φ=ℤ/5ℤ, so ℤ/5ℤ is isomorphic to the quotient by the kernel.
02 / Prove
Why is a kernel normal?
Hint
Apply φ to gng⁻¹ for n in the kernel.
Solution
φ(gng⁻¹)=φ(g)eφ(g)⁻¹=e, so conjugation preserves the kernel. Applying the same reasoning with g⁻¹ gives equality.
03 / Connect
For a linear map T: V→W, explain why V/ker T is naturally isomorphic to im T.
Hint
Send the coset v+ker T to T(v).
Solution
If v−v′∈ker T then T(v)=T(v′), so the map is well-defined. It is linear and onto im T. Its kernel contains only the zero coset, so it is an isomorphism. In finite dimension, taking dimensions recovers rank–nullity.
Connect the ideas
Build on this
These sheets use this topic as a prerequisite.
Go deeper
Further reading and full-length treatments: