Group Theory - The Basics with Examples

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A group − denoted (G,*) − is a set G of elements together with an operation (*) on pairs of elements x,y∈G such that four defining properties are fulfilled:

Closure:
∀x,y∈G | x∗y∈G

Associativity:
∀x,y,z∈G | (x∗y)∗z=x∗(y∗z)

An "Identity" Element Exists:
∃e∈G | e∗x=x∗e=x

An "Inverse" Element Exists:
∀x∈G, ∃y∈G | x∗y=y∗x=e

Examples

Q1. Show that (Z,+) is a group.

Q2. Show that (Z,×) is not a group.

Q3. Show that (Q∗,×) is a group.

Q4. Show that (Q,×) is not a group.

Q5. If determine if (N,+) is a group.

Q6. If P represents the set of all prime numbers, determine if (P,×) is a group.

Q7. Show that (Z_5,+) forms a group.