Which of the following is NOT a group under the given operation?
A(Z, +): integers under addition
B(R\{0}, ×): nonzero reals under multiplication
C(Z, ×): integers under multiplication
D(Z/7Z, +): integers mod 7 under addition
Integers under multiplication fail the inverse axiom: the element 2 has no multiplicative inverse in Z (1/2 is not an integer). The other three sets satisfy all four group axioms: closure, associativity, identity, and inverses.
Question 2 True / False
Nearly every group is commutative — that is, a * b = b * a for most elements a and b.
TTrue
FFalse
Answer: False
Groups need not be commutative. The symmetric group S_3 (permutations of 3 elements) is a standard counterexample: composing two permutations in different orders typically gives different results. Groups where commutativity holds are called abelian; non-abelian groups are common and important.
Question 3 Short Answer
In the group (Z/4Z, +) — integers mod 4 under addition — what is the inverse of the element 3?
Think about your answer, then reveal below.
Model answer: 1, because 3 + 1 = 4 ≡ 0 (mod 4), the identity element.
The inverse of an element a is whatever element b satisfies a + b = e, where e is the identity. Here e = 0, and 3 + 1 = 4 = 0 mod 4, so the inverse of 3 is 1. This illustrates that the inverse is not always the additive negative in the usual integer sense.