Question

Is {1, 2, 3,4) a group under .5 (multiplication modulo 5)? If so, is it isomorphic to 24 or to Z2 x Z2?

Answer 1

Answer with step by step explanation:

We are given that a set

Let A={1,2,3,4}

We have to prove that it is a group under modulo 5

`a*_n b}=(a\cdot b)/(5)=remainder`

Closed property:
`a*_5 b=(a* b)/(5)=remainder \in A`for all a,b belongs to A

Associative property:It is satisfied property

`a*_5(b*_5c)=(a*_5b)*_5c` for all a,b,c belongs to A

Identity :
`a*_5e=a` Where e is identity of group

`2*_51=2,3*_5 1=3,4*_51=4`

Hence, identity exist ,e=1

Inverse:
`a*_5a^(-1)=e`

`2*_53=1,3*_52=1,4*_4=1,1*_5=1`

Hence, inverse exist of every element

Given set satisfied all properties of group under multiplication modulo.Therefore, givens set is a group under multiplication modulo.

It is a U(5) because a group under multiplication modulo is called U(n) group U(n)={r, gcd(r,n)=1}

We know that order of group U(n)=
`\phi(n)`

Order of U(5)=4

We know that
`U(p^n)` isomorphic to
`Z_(p^n-p^(n-1))`

p=5,n=1

U(5)isomorphic to
`Z_(5-1)=Z_4`