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Exercise 15.2.17. Use Exercise 15.2.16 to prove the following proposition: Proposition 15.2.18. The identity element in a group G is unique; that is, there exists only one elemente € G such that eg = ge=g for all g = G.

Exercise 15.2.17. Use Exercise 15.2.16 to prove the following proposition: Proposition 15.2.18. The identity element in a group G is unique; that is, there exists only one elemente € G such that eg = ge=g for all g = G.

Elements Of Modern Algebra
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 32E: (See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup...
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Please do Exercise 15.2.17. Please show step by step and explain

Hint I got for that questions is "Suppose that e and f are both identities of G, and use
Exercise 15.2.16 to show that this implies e = f"

The next three exercises are very useful in helping determine whether or not a given Cayley table represents a group. Exercise 15.2.16. Given h is an element of (G, o). 1. Show that h is an identity element of G if and only if there exists a g € G such that hog = g. (*Hint*) 2. Show that h is an identity element of G if and only if there exists a g €G such that goh = g. In Exercise 15.2.16 we were careful to say an identity element. Could a group have multiple identity elements? Let's settle the question once and for all: Exercise 15.2.17. Use Exercise 15.2.16 to prove the following proposition: Proposition 15.2.18. The identity element in a group G is unique; that is, there exists only one element e G such that eg = ge=g for all g € G.
Transcribed Image Text:The next three exercises are very useful in helping determine whether or not a given Cayley table represents a group. Exercise 15.2.16. Given h is an element of (G, o). 1. Show that h is an identity element of G if and only if there exists a g € G such that hog = g. (*Hint*) 2. Show that h is an identity element of G if and only if there exists a g €G such that goh = g. In Exercise 15.2.16 we were careful to say an identity element. Could a group have multiple identity elements? Let's settle the question once and for all: Exercise 15.2.17. Use Exercise 15.2.16 to prove the following proposition: Proposition 15.2.18. The identity element in a group G is unique; that is, there exists only one element e G such that eg = ge=g for all g € G.
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