MATHS 302: ALGEBRA TEST QUESTIONS AND ANSWERS A+

GROUP DEFINITIONS GROUPOID: Is a mathematical system consisting of a non- empty set S of arbitrary elements together with a single binary operation defined on S. It is denoted by an ordered pair (S, *). Note: A group is determined not by its set of elements alone, but by the set together with the binary operation defined on it. Different groupoids may have the same associated set of elements, and this is the reason for representing a groupoidasan ordered pair (S, *) of set Sand binary operation * defined onS. SEMI- GROUP: A semi- gr oup is a gr oupoid (S, o) whose binar y oper ation “ o” satisfies closure and associative law. MONOID: It’s a semi- gr oup with identity. That’s if e ∈ S, we have a oe = e oa = a for all a ∈ S. GROUP: A pair (G, o) is a gr oup if, and only if (G, o) is a semi gr oup with an identity in which each element of G has an inver se in G. The definition of a gr oup is redefined as. Given (G, * ) as an algebraic str uctur e wher e Gis a non- empty set with a binary oper ation * . Then (G, * ) is called a group if the following axioms hold: C1: Gisclosed withrespect to* i:e ∀ x,y ϵ Git implies (x*y)ϵ G C2: * is associative in G i:e ∀ x,y,z ϵ Git implies (x*y) * z = x *(y * z) C3: ∃ an identity element e say in Gw.r.t .i:e e ϵ Gsuchthat ∀ gϵG: e * g = g* e = g C4: ∀ gϵG, ∃ g - 1 ϵG suchthat g* g - 1 = g- 1 * g = eϵG where e is the identity in G. If C1- C4 holdsinG, then it impliesGisa group. REMARKS • If there exists an element e of a group (G,o) such that e1 o a = a for every a of G, e1 is called a left identity of (G, o). • If there exists an element e of a group (G,o) such that a o e = a for every a of G, e is called a right identity of (G, o). • If a gr oup (G,o) has both a left and r ight identity, then these two are equal. (This element which is both the left and r ight identity of (G ,o) is called the identity element of the gr oup (G,o). • If for a given a ∈ G, ther e exists an element a - 1 ∈ G of the gr oup (G ,o) with an identity e such that a - 1oa = e, a- 1 is called the left inverse of a. • If for a given a ∈ G, ther e exists an element a - 1 ∈ G of the gr oup (G ,o) with an identity e such that a oa- 1 = e, a - 1 is called the right inverse of a. • If a gr oup (G,o) has both a left and r ight inver se of an element a, then both these inver ses are equal. (This element which is both the left and r ight inver se of (G,o) is called the inverse element of a