Question

How many subgroups does the group `(ZZ_5, o+_5)` have ?

Answer 1

`2`

Explanation:

`(ZZ_5, o+_5)` is the set of integers modulo `5` with addition modulo `5`.

That is "clock addition" modulo `5`.

Since `5` is prime, there are only two subsets which are closed under addition, namely the whole set `{0, 1, 2, 3, 4}` and the set containing only `{0}`.

In other words, the only two subgroups are:

`(ZZ_5, o+_5)` and `({0}, +)`

Here's the addition table for the group `(ZZ_5, o+_5)`:

`underline(color(white)(0)o+_5|color(white)(0)0color(white)(00)1color(white)(00)2color(white)(00)3color(white)(00)4color(white)(0))`
`color(white)(o+_5)0|color(white)(0)0color(white)(00)1color(white)(00)2color(white)(00)3color(white)(00)4`
`color(white)(o+_5)1|color(white)(0)1color(white)(00)2color(white)(00)3color(white)(00)4color(white)(00)0`
`color(white)(o+_5)2|color(white)(0)2color(white)(00)3color(white)(00)4color(white)(00)0color(white)(00)1`
`color(white)(o+_5)3|color(white)(0)3color(white)(00)4color(white)(00)0color(white)(00)1color(white)(00)2`
`color(white)(o+_5)4|color(white)(0)4color(white)(00)0color(white)(00)1color(white)(00)2color(white)(00)3`