Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 9 / a

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- Solution a

Solution

A group homomorphism

f : G \to G^{'}

satisfies

f (g h) = f (g) f (h)

for all

g, h \in G

. It consequently sends the identity element to the identity and inverses to inverses. Its group homomorphism kernel is

ker f = {g \in G : f (g) = e_{G^{'}}} .

A normal subgroup

H

of

G

is a subgroup invariant under conjugation:

g H g^{- 1} = H for every g \in G .

Equivalently,

g H = H g

for every

g

, so left and right cosets coincide.

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