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$\mathbb Z$ (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.

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    $\begingroup$ The order of a group is its cardinality, see here. $\endgroup$ – Dietrich Burde Mar 16 '15 at 21:15
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    $\begingroup$ To give a fish, or to teach to fish? If you want to figure out what the definition of order is, you can simply google order group. Search engines are very useful! $\endgroup$ – whacka Mar 16 '15 at 21:17
  • $\begingroup$ So the Z_2 means order 2 and the Z_4 means order 4? $\endgroup$ – user224139 Mar 16 '15 at 21:18
  • $\begingroup$ Its not so much order I wanted to look up. It's the notation of "Z_2". I read that on wikipedia too, but there were two definitions for it so I wasn't too sure. But Hey! You couldn't fault me for trying to get information easier, can ya? :) $\endgroup$ – user224139 Mar 16 '15 at 21:19
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    $\begingroup$ Correct, there are (at least) two definitions. Here, $\mathbb Z_n$ is the cyclic group of order $n$. But one also may see $\mathbb Z_p$ for the ring of $p$-adic integers. $\endgroup$ – GEdgar Mar 16 '15 at 21:27
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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.)

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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.

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