Please do Exercise 15.2.17. Please show step by step and explainHint I got for that questions is "Suppose that e and f are both identities of G, and useExercise 15.2.16 to show that this implies e = f" The next three exercises are very useful in helping determine whether ornot 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 ag € G such that hog = g. (*Hint*)2. Show that h is an identity element of G if and only if there exists ag €G such that goh = g.In Exercise 15.2.16 we were careful to say an identity element. Could agroup have multiple identity elements? Let's settle the question once andfor 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; thatis, there exists only one element e G such that eg = ge=g for all g € G.

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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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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.

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