Abstract Algebra: Theory and Applications · Problem 10.4.1

Homomorphisms — Thomas Judson

Mathematics — Abstract Algebra · normal-subgroups-factor-groups

Problem

For each of the following groups

G

, determine whether

H

is a normal subgroup of

G

. If

H

is a normal subgroup, write out a Cayley table for the factor group

G/H

. (a)

G = S_4

and

H = A_4

(b)

G = A_5

and

H = \{(1),(1\,2\,3),(1\,3\,2)\}

(c)

G = S_4

and

H = D_4

(d)

G = Q_8

and

H = \{1, -1, I, -I\}

(e)

G = \mathbb{Z}

and

H = 5\mathbb{Z}

Hint. (a)

\begin{array}{c|cc} & A_4 & (1\,2)A_4 \\ \hline A_4 & A_4 & (1\,2)A_4 \\ (1\,2)A_4 & (1\,2)A_4 & A_4 \end{array}

(c)

D_4

is not normal in

S_4