(b) Let T1 = {o E Sn : 0(1) = 1}, with (n > 1).(i) Prove that T, is a subgroup of Sn, andhence, deduce that S, has a subgroup of order (n – 1)!(ii) Show that T, = {o E Sn : 0(1) = 2}, is a left coset of the subgroup T,.(iii) List the set of all distinct left cosets and right cosets of the subgroup T,, when n = 4.For each such coset, give its members explicitly.(iv) Determine whether T, is a normal subgroup or not.

“Since you have asked multiple questions, we will solve the first question for you. If you want any…

Answered: Let T; = {o € S, : 0(1) = 1}, with (n >…

Let T; = {o € S, : 0(1) = 1}, with (n > 1). Prove that T, is a subgroup of S,, and hence, deduce that S„ has a subgroup of order (n – 1)!

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 2E: For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H...

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Question

(b) Let T1 = {o E Sn : 0(1) = 1}, with (n > 1).
(i) Prove that T, is a subgroup of Sn, and
hence, deduce that S, has a subgroup of order (n – 1)!
(ii) Show that T, = {o E Sn : 0(1) = 2}, is a left coset of the subgroup T,.
(iii) List the set of all distinct left cosets and right cosets of the subgroup T,, when n = 4.
For each such coset, give its members explicitly.
(iv) Determine whether T, is a normal subgroup or not.

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