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# How to check if $\mathbb{Z}_n\setminus \{\bar{0}\}$ create a group (or not a group) with multiplication ? (with given n)

How to check if $\mathbb{Z}_n\setminus \{\bar{0}\} $ create a group with multiplication ? (with given n)

For example, if $\mathbb{Z}_{41}$ create a group with multiplication.

I aware that I need to prove 4 things:

Closure:

For all a, b in G, the result of the operation, a • b, is also in G.b[›]

Associativity:

For all a, b and c in G, (a • b) • c = a • (b • c).

Identity element:

There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique (see below), and thus one speaks of the identity element.

Inverse element:

For each a in G, there exists an element b in G, commonly denoted $a^{-1}$, such that a • b = b • a = e, where e is the identity element.

But I have trouble applying modulo in this case. Please give me a formal example.