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.
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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