Question

Lower LFT Group?:

(a) Consider a set G with a product G×G→G:(g₁,g₂)→g₁g₂.
What properties must G satisfy in order to be called a group?

Answer 1

A set G with a product operation qualifies as a group if it satisfies closure, associativity, has an identity element, and every element has an inverse. Commutativity is not a requirement for a set to be a group.

Step-by-step explanation:

To be called a group, a set G with a product operation G×G→G must satisfy four fundamental properties:

1. Closure: For all elements g1 and g2 in G, the product g1g2 must also be in G.
2. Associativity: The product operation must be associative, meaning that for all elements g1, g2, and g3 in G, the relation (g1g2)g3 = g1(g2g3) holds.
3. Existence of identity element: There exists an element e in G such that for every element g in G, the equation eg = ge = g holds.
4. Existence of inverse elements: For every element g in G, there exists an element g-1 in G such that gg-1 = g-1g = e, where e is the identity element.

Note that the commutative property is not required for a set to be a group. Some groups are commutative (or abelian), meaning that g1g2 = g2g1 for all g1, g2 in G, but it is not a necessary condition.