Abstract Algebra: Theory and Applications · Problem 10.4.13

Homomorphisms — Thomas Judson

Mathematics — Abstract Algebra · center-normal-subgroups

Problem

Recall that the center of a group

G

is the set

Z(G) = \{x \in G : xg = gx \text{ for all } g \in G\}.

(a) Calculate the center of

S_3

. (b) Calculate the center of

GL_2(\mathbb{R})

. (c) Show that the center of any group

G

is a normal subgroup of

G

. (d) If

G/Z(G)

is cyclic, show that

G

is abelian.