What does $3\mathbb{Z}/12\mathbb{Z}$ mean?

Question

What does $3\mathbb{Z}/12\mathbb{Z}$ mean? I must find all of its elements.

Thank you

Answer 1

Letting $\Bbb Z/12\Bbb Z=\{\overline0,\overline1,\ldots,\overline{11}\}$. $3\Bbb Z/12\Bbb Z$ would be $\{\overline0,\overline3,\overline6,\overline9\}$.

Answer 2

It's what's called a quotient group, you may wish to look them up online.

Answer 3

$3 \mathbb Z$ is the group generated by $3$ under addition or $<3,+>$ namely $\{...-3,0,3,6...\}$

$12\mathbb Z$ is the group generated by $12$ under addition or $<12,+>$ namely $\{...-12,0,12..\}$

Clearly every multiple of $12$ is a multiple of $3$. In fact $<12>$ is a subgroup of $<3>$

We can compute the left cosets of $<12>$

$3+<12>=\{...-9,3,15..\}$

$6+<12>=\{...-6,6,18...\}$

$9+<12>=\{...-3,9,21...\}$

Notice that they are shifted multiples of $12$ so eventually

$12+<12>=\{...-12,0,12...\}$ I.e the original subgroup.

It's easy to see the right cosets are the same and so $<12>$ is a normal subgroup.

You use the cosets as elements of a group themselves. Essentially you pick an element from each coset you're adding and find out which cosets the answer is in.

This group is denoted $3\mathbb Z /12\mathbb Z$