Please do Exercise 18.2.2 part B and C and please show step by step and explain Proposition 18.2.1. Let H be a subgroup of a group G and suppose that91,92 € G. The following conditions are equivalent.1. 91H = 92H;2.9₁ ¹92 € H.3. 92 € 9₁H;4. 92H C 9₁H;5. Hg₁¹ = Hg₂¹;(Note: "C" means that equality is also possible)The proof of this Proposition is laid out in the Exercise 18.2.2 below,and you are asked to fill in the details. Parts (a)-(f) of the exercise establishthe following steps:(1) (2) (³) (4) (¹) and (2)(e.f)Exercise 18.2.2.(a) Show that condition (1) implies condition (2). (*Hint*)(b) Show that condition (2) implies condition (3). (*Hint*)(c) Show that condition (3) implies condition (4). (*Hint*)(d) Show that condition (4) implies condition (1). (*Hint*)(e) Show that condition (2) implies condition (5). (*Hint*)(f) Show that condition (5) implies condition (2).(5).

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Answered: (b) Show that condition (2) implies…

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(b) Show that condition (2) implies condition (3). (c) Show that condition (3) implies condition (4).

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 19E: Find the order of each of the following elements in the multiplicative group of units . for for ...

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Question

Please do Exercise 18.2.2 part B and C and please show step by step and explain

Proposition 18.2.1. Let H be a subgroup of a group G and suppose that
91,92 € G. The following conditions are equivalent.
1. 91H = 92H;
2.9₁ ¹92 € H.
3. 92 € 9₁H;
4. 92H C 9₁H;
5. Hg₁¹ = Hg₂¹;
(Note: "C" means that equality is also possible)
The proof of this Proposition is laid out in the Exercise 18.2.2 below,
and you are asked to fill in the details. Parts (a)-(f) of the exercise establish
the following steps:
(1) (2) (³) (4) (¹) and (2)
(e.f)
Exercise 18.2.2.
(a) Show that condition (1) implies condition (2). (*Hint*)
(b) Show that condition (2) implies condition (3). (*Hint*)
(c) Show that condition (3) implies condition (4). (*Hint*)
(d) Show that condition (4) implies condition (1). (*Hint*)
(e) Show that condition (2) implies condition (5). (*Hint*)
(f) Show that condition (5) implies condition (2).
(5).

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