Abstract Algebra: Theory and Applications · Problem 13.9

Group Actions — Thomas Judson

Mathematics — Abstract Algebra · composition-series

Problem

Find all of the composition series for each of the following groups. ((a))

\mathbb{Z}_{12}

((b))

\mathbb{Z}_{48}

((c)) The quaternions,

Q_8

((d))

D_4

((e))

S_3 \times \mathbb{Z}_4

((f))

S_4

((g))

S_n

n \geq 5

((h))

\mathbb{Q}

Hint. (a)

\{0\} \subset \langle 6 \rangle \subset \langle 3 \rangle \subset \mathbb{Z}_{12}

; (e)

\{(1)\}\times\{0\} \subset \{(1),(1\,2\,3),(1\,3\,2)\}\times\{0\} \subset S_3 \times \{0\} \subset S_3 \times \langle 2 \rangle \subset S_3 \times \mathbb{Z}_4