Let G be a group and let a e G. The set C(a) = {x € G | xa = that commute with a is called the Centralizer of a in G. The set Z(G) = {x E G| xa = ax for all a e G} of all elements of G that commute with every element of G is called the Center of G. ax} of all elements (a) Prove that Z(G) is a subgroup of G. (b) Prove that Cg(a) is a subgroup of G. (c) Compute CG(a) when G = S3 and a = (d) Compute CG(a) when G = (e) Prove that Z(G) = NaegCg(a). (1,2). Są and a = (1, 2).
Let G be a group and let a e G. The set C(a) = {x € G | xa = that commute with a is called the Centralizer of a in G. The set Z(G) = {x E G| xa = ax for all a e G} of all elements of G that commute with every element of G is called the Center of G. ax} of all elements (a) Prove that Z(G) is a subgroup of G. (b) Prove that Cg(a) is a subgroup of G. (c) Compute CG(a) when G = S3 and a = (d) Compute CG(a) when G = (e) Prove that Z(G) = NaegCg(a). (1,2). Są and a = (1, 2).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please just answer d & e. Thank you
Transcribed Image Text:Let G be a group and let a e G. The set C(a) = {x € G | xa =
that commute with a is called the Centralizer of a in G. The set Z(G) = {x E G| xa =
ax for all a e G} of all elements of G that commute with every element of G is called the
Center of G.
ax} of all elements
(a) Prove that Z(G) is a subgroup of G.
(b) Prove that Cg(a) is a subgroup of G.
(c) Compute CG(a) when G = S3 and a =
(1,2).
(d) Compute Cg(a) when G = Są and a =
(1, 2).
(e) Prove that Z(G) = NaegCg(a).
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