Let be a positive integer and

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

Let

n

be a positive integer and

\mathbb{Z}_n = \{0,1,2,\dots,n-1\}

, with

a\oplus b

defined as the remainder when the ordinary integer sum

a+b

is divided by

n

. ((a)) Prove that

(\mathbb{Z}_n,\oplus)

satisfies the four group axioms for every

n\ge1

: closure, associativity, an identity element and an inverse for each element. State the inverse of a general element

a

in closed form. ((b)) For

n=5

, write out the

5\times5

addition table and list, in order, the inverse of each of

0,1,2,3,4