Question

Consider the integers $\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}$ under addition (mod 4). Write the group "multiplication" table and show that you have a group of order 4 . Is this group isomorphic to the cyclic group of order 4 or to the${\mathbf{4}}^{\mathbf{3}}$group?

Short answer

Yes, this is isomorphic to 4's group.

Step-by-step solution

Given information

Show that with the multiplication table that the integers$0,1,2,3$ with addition (mod 4 is a group. Also, is this a group isomorphic to the cyclic group of order 4 or to the 4’s group

Definition of Group

A group is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associatively, the identity property, and the inverse property.

Verify the given statement

To verify that it is a group, we first note that 0 is the unit element in addition.

Now will show the closure property

$\begin{array}{l}1+1=2\\ 1+2=3\\ 1+3=0\\ 2+2=0\end{array}$

$\begin{array}{l}2+3=1\\ 3+3=2\end{array}$

The output result shown that 1 is the inverse of 3 and vice versa, and 0 and 2 are its own inverses. Number addition is associative, so this is a group. The multiplication table is given as,

$\begin{array}{cccc}0& 1& 2& 3\\ 0& 1& 2& 3\\ 1& 2& 3& 0\\ 2& 3& 0& 1\\ 3& 0& 1& 2\end{array}$

This is isomorphic to the cyclic group 4's.