← MATH 01340 · Modern Algebra I
MATH 01340 · Groups
Group axioms
Definition
A group (G,∗) is a set with associative operation, identity e, and every element invertible. Abelian if commutative. Classic examples: (ℤ,+), (ℤₙ,+), U(n) units mod n, Sₙ permutations.
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Check all axioms when proving something is a group — associativity is often the subtle one for new operations.
Cancellation laws follow from inverses.
Non-examples teach as much as examples (natural numbers under subtraction fail).
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Attempt each problem first, then open the worked solution.
1.Prove ℤ₅ under addition mod 5 is a group.