Is Z10 Z12 Z6 Z60 Z6 Z2 ProofSolutionsolution There is no su

Is Z₁₀ Z₁₂ Z₆ Z₆₀ Z₆ Z₂? Proof

Solution

solution

There is no such thing as \"approximately equal to\" for groups.

Clearly, it is just typographic differences. The two groups are congruent / isomorphic. It is asking you to prove this statement.

Zn is the group of integers modulo n.

The simplest way to prove they are congruent is to use three important theorems.

Theorem 1: Zm Zn Zn Zm

Proof: (a,b) = (b,a) is an isomorphism.

Theorem 2: Zm Zn Z(nm) for gcd(m,n)=1

Proof: (a,b) = na+mb is an isomorphism.

Theorem 3: ( Zm Zn ) Zk Zm (Zn Zk )

Proof: ( (a,b),c ) = (a,(b,c)) is an isomorphism.

So you can \"break down\" each factor into prime components (by theorem 2) and then rearrange them (by theorem 1). Theorem 3 tells us parentheses are irrelevant -- the statement of your theorem already excludes parentheses.

You conclude:

Z10 Z12 Z6 ( Z2 Z5 ) ( Z3 Z4 ) ( Z2 Z3 )

Z2 Z2 Z3 Z3 Z4 Z5

Z60 Z6 Z2 ( Z4 Z3 Z5 ) ( Z2 Z3 ) Z2

Z2 Z2 Z3 Z3 Z4 Z5


Since the two results are equal, by transitivity of we get QED.