Abstract Algebra: Theory and Applications · Problem 9.4.45

Normal Subgroups and Factor Groups — Thomas Judson

Mathematics — Abstract Algebra · internal-direct-products

Problem

Let

G

be the internal direct product of subgroups

H

and

K

. Show that the map

\phi : G \to H \times K

defined by

\phi(g) = (h, k)

for

g = hk

, where

h \in H

and

k \in K

, is one-to-one and onto. Hint. To show that

\phi

is one-to-one, let

g_1 = h_1k_1

and

g_2 = h_2k_2

and consider

\phi(g_1) = \phi(g_2)