What does this notation mean: $\mathbb{Z}_2$

Question

$\mathbb Z$ (Our usual notation for the integers) with a little subscript at the bottom.

This is the question being asked:

what are the subgroups of order $4$ of $\mathbb Z_2 \times\mathbb Z_4$ ($\mathbb Z_2$ cross $\mathbb Z_4$)

Give them as sets and identity the group of order 4 that each of the subgroup is isomorphic to

I was thinking that it meant the set of integers modulo $4$ and modulo $2$, but I'm not too sure

Give them as sets and identity the group of order $4$ that each of the subgroup is isomorphic to

What is the definition of "order". I couldn't really find that anywhere either.

Answer 1

In the context above, $\mathbb{Z}_n=\{0,1,\ldots, n-1\}$, where $n\in\mathbb{N}$.

This is the group of integers modulo $n$. (It is a group under addition modulo $n$.)

So $\mathbb{Z}_2=\{0,1\}$, the group of integers modulo $2$.

The order of a group is its cardinality.

(Just in case you're interested, the order of an element of a group is the smallest $n\in\mathbb{N}:a^n=e$, where $e$ is the identity and $a$ is an element of the group. If no such $n$ exists, then $a$ is said to have infinite order.)

Answer 2

There is some ambiguity here. My text, Dummit and Foote, insists that $\mathbb Z_p$ is the cyclic group of order p, not the ring of integers modp. On this site, most often, $\mathbb Z_p$ refers to the ring of integers modp, and $C_p$ is the typical symbol for the cyclic group of order p. If you know, as in this case, that you are dealing with a group rather than a ring, the meaning is clear.