 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/8/b/solution

H \leq G

and

g \in G

, the left and right cosets are

g H = {g h : h \in H}

and

H g = {h g : h \in H}

g H g^{- 1} = H

for every

g

. Multiplying this set equality on the right by

g

gives

g H = H g for every g \in G .

Conversely this equality implies

g H g^{- 1} = H

, so it is an equivalent definition of normality.

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