GATE CS 2009 Q1 — Monoids and Groups

By definition, an algebraic structure $(G, *)$ is a group if it satisfies the following four axioms:

1. Closure: For all $a, b \in G$, $a * b \in G$.
2. Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$.
3. Existence of identity: There exists an element $e \in G$ such that $a * e = e * a = a$ for all $a \in G$.
4. Existence of inverse: For every $a \in G$, there exists an element $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$.

Commutativity (i.e., $a * b = b * a$ for all $a, b \in G$) is an additional property. Groups that satisfy commutativity are called Abelian groups. Many groups are non-Abelian (such as the group of invertible $n \times n$ matrices under matrix multiplication, or the symmetric group $S_3$).

Thus, commutativity is NOT necessarily a property of a group.