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(6) Define the center of a group G as the subset Z = {z EG: zg = gz for all g € G}. Prove that Z is a normal subgroup of G.

(6) Define the center of a group G as the subset Z = {z EG: zg = gz for all g € G}. Prove that Z is a normal subgroup of G.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Abstract Algebra I 

1. Show that \( S_4 \) does not contain a normal subgroup of order 2. 2. Define the *center* of a group \( G \) as the subset \[ Z = \{ z \in G : zg = gz \text{ for all } g \in G \}. \] Prove that \( Z \) is a normal subgroup of \( G \).
Transcribed Image Text:1. Show that \( S_4 \) does not contain a normal subgroup of order 2. 2. Define the *center* of a group \( G \) as the subset \[ Z = \{ z \in G : zg = gz \text{ for all } g \in G \}. \] Prove that \( Z \) is a normal subgroup of \( G \).
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