# 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.

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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.

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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.

Step 3

Proof of ...

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