Abstract Algebra: Theory and Applications · Problem 6.5.5

Introduction to Cryptography — Thomas Judson

Mathematics — Abstract Algebra · cosets-lagrange-theorem

Problem

List the left and right cosets of the subgroups in each of the following. ((a))

\langle 8 \rangle

\mathbb{Z}_{24}

((b))

\langle 3 \rangle

U(8)

((c))

3\mathbb{Z}

\mathbb{Z}

((d))

A_4

S_4

((e))

A_n

S_n

((f))

D_4

S_4

((g))

\mathbb{T}

\mathbb{C}^*

((h))

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

S_4

Hint. (a)

\langle 8\rangle

1+\langle 8\rangle

2+\langle 8\rangle

3+\langle 8\rangle

4+\langle 8\rangle

5+\langle 8\rangle

6+\langle 8\rangle

, and

7+\langle 8\rangle

; (c)

3\mathbb{Z}

1+3\mathbb{Z}

, and

2+3\mathbb{Z}