Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 9 / a

Write

H = {e, h}

with

h \neq = e

. For any

g \in G

, the normal subgroup property implies

g h g^{- 1} \in H

. This conjugate cannot be

e

g^{- 1}

and

g

would then give

h = e

. There is only one remaining choice, so

g h g^{- 1} = h, g h = h g .

Thus

h

commutes with every element of

G

; the identity does too. Hence

H \subseteq Z (G) .

This proves that a normal subgroup of order two is central. The proof does not require

G

to be finite: conjugation permutes

H

and fixes its identity, so it must fix its only other element.

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