Abstract Algebra: Theory and Applications · Problem 6.5.11

Introduction to Cryptography — Thomas Judson

Mathematics — Abstract Algebra · cosets-lagrange-theorem

Problem

Let

H

be a subgroup of a group

G

and suppose that

g_1, g_2 \in G

. Prove that the following conditions are equivalent. ((a))

g_1 H = g_2 H

((b))

H g_1^{-1} = H g_2^{-1}

((c))

g_1 H \subset g_2 H

((d))

g_2 \in g_1 H

((e))

g_1^{-1} g_2 \in H