= = Definition 1.4. Let G be a group the set Z(G) = {x E G : gx = xg for all g € G} is the G E center of G. Problem 1.7.

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Definition 1 4 Let G Be A Group The Set Z G X E G Gx Xg For All G G Is The G E Center Of G Problem 1 7 1 (99.11 KiB) Viewed 12 times

= = Definition 1.4. Let G be a group the set Z(G) = {x E G : gx = xg for all g € G} is the G E center of G. Problem 1.7. 1. Show that Z(G) is a subgroup of G. ጎ = 2. Recall that for an integer n > 3 we have the dihedral group Dn = {e, a, a’, ...an–1, 6, ba, baʼ, ... ba"–1} , b, with product determined by the relations a" = e, 62 = e and bak = a-kb. What is ZDn). Prove your answer. Hint: The answer will depend on the parity of n. = a