Q1-(Uniqueness of the inverse:). Show that for every a ∈ G, the inverse of a is unique. We will denote the inverse of a by a−1. Also prove the following:(1) (a−1)−1 =a for all a ∈G.
(2) (a · b)−1 = b−1 · a−1 for all a, b ∈ G.(3) (b · a)−1 = a−1 · b−1 for all a, b ∈ G. Q2- Let S be a set on which an associative binary operation · has been defined such that S contains an identity element eS. Let U(S) be the set of all the units in S. Prove that U(S) is a group with respect to the operation ·. It is called the group of units of S.  Q3- (Subgroup Test for Finite Groups). Let G be a finite group. Prove that a nonempty subset H ⊆ G is a subgroup of G if and only if H is closed under the group operation of G.

Question
Asked Apr 20, 2019
21 views

Q1-(Uniqueness of the inverse:). Show that for every a ∈ G, the inverse of a is unique. We will denote the inverse of a by a−1. Also prove the following:

(1) (a−1)−1 =a for all a ∈G.


(2) (a · b)−1 = b−1 · a−1 for all a, b ∈ G.

(3) (b · a)−1 = a−1 · b−1 for all a, b ∈ G.

 

Q2- Let S be a set on which an associative binary operation · has been defined such that S contains an identity element eS. Let U(S) be the set of all the units in S. Prove that U(S) is a group with respect to the operation ·. It is called the group of units of S.

 

 

Q3- (Subgroup Test for Finite Groups). Let G be a finite group. Prove that a nonempty subset H ⊆ G is a subgroup of G if and only if H is closed under the group operation of G.

check_circle

Expert Answer

Step 1

As per norms, the first question Q1 (with 3 subparts ) is answered. To prove the stated properties of the inverese operation in a group.

Step 2

By definition, every element a in a group has an inverse element a^-1 defined as shown. We first show that the inverse of an element is unique.

fullscreen
Step 3

Proof of ...

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in

Math

Advanced Math

Related Advanced Math Q&A

Find answers to questions asked by student like you
Show more Q&A
add
question_answer

Q: Need help with these 4 problems.

A: Hey, since there are multiple questions posted, we will answer first question. If you want any speci...

question_answer

Q: Jose received $450 for his birthday from his family.  He wishes to buy a motorcycle and decides to u...

A: To describe the required sets in the three set theory problems

question_answer

Q: Given the differential equation: (1-x)y"+y=0, x0=0 Find: Seek the power series solution for the dif...

A: Consider the differential equation

question_answer

Q: Convert the number to decimal form. 251seven

A: To find the base 10 representation for the number given its base 7 representation.

question_answer

Q: the book is "First Course on Fuzzy Theory and Applications"

A: To prove that the given fuzzy set is convex and to determine the α-cut sets for the given values of ...

question_answer

Q: 5.2.6.B       Discrete Mathematics This is a dublicate please answer the question as quickly as poss...

A: Known fact:Base step: Show it is true for first case, usually n =1.Induction step: Show that if n = ...

question_answer

Q: I need help with this problem Please. Thanks!

A: To convert the word problem into inequalities and represent them graphically

question_answer

Q: please explain a and b to me step by step and dont skip any steps. pls write clearly and  i will rat...

A: (a)Consider the given matrix:

question_answer

Q: Let z be a complex variable  z=x+iy question attached in pic

A: To examine the complex integral of the given rational function over different contours (circles) in ...