$\mathbb Z$ (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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$\begingroup$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.)
$\endgroup$ 1 $\begingroup$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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