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What is the difference between these notations $\mathbf{\mathbb{Z}_p^{*}}$ and $\mathbf{(\mathbb{Z}/p\mathbb{Z})^{*}}$ in terms of group theory in abstract algebra?

and can anyone give me an example of multiplicative group of non-prime order?

Thanks!

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    The multiplicative groups you have up there have order $p-1,$ which is almost never a prime... – 2017-01-17
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    What do you mean by $\mathbb Z_p^*$? The notation $\mathbb Z_n$ can mean one of a few things (The cyclic group of order $n$, sometimes the ring $\mathbb Z/n\mathbb Z$, sometimes the $n$-adic numbers). Depending on that answer, you'll have a different potential relationship. – 2017-01-17
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    @Mark $Z_p^{*}$ multiplicative group of modulo p. – 2017-01-17
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    @spaceisdarkgreen I was trying to say can we have an example of multiplicative group other than $Z_p^{*}$. – 2017-01-17
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    But that is not what you said. You asked for a multiplicative group of nonprime order, and $\Bbb Z_p^\ast$ is a multiplicative group of nonprime order. Or your could take $\Bbb Z_n^\ast$ for other natural numbers $n$, or you could take the $m$th roots of unity in $\Bbb C$ where $m$ is not prime, or you could take positive reals, etc. – 2017-01-18
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    Anyway, $\Bbb Z_n$ is typically shorthand for $\Bbb Z/n\Bbb Z$, but is sometimes defined more directly without discussing quotient rings. (Both could refer to either the additive group of the ring of integers mod $n$.) – 2017-01-18

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