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Formal Definition

A group $\langle G,*\rangle$ is a set $G$,closed under a binary operation $*$,such that the following axioms are satisfied:

$\mathscr{Y}_1$:For all $a,b,c\in G$,we have $(a*b)*c=a*(b*c)$. (associativity of *)

$\mathscr{Y}_2$:There is an element $e$ in $G$ such that for all $x\in G$,$e*x=x*e=x$. ($e$ is an identity element for $*$.)

$\mathscr{Y}_3$:Corresponding to each $a\in G$,there is an element $a'$ in $G$ such that $a*a'=a'*a=e$. ($a'$ is an inverse of $a$.)

Informal Definition

A group is a set with a binary operation that satisfies these axioms: it must be closed under the binary operation, associative, contain the identity element, and each element in the set has a corresponding inverse.


$\langle U,\cdot\rangle$ and $\langle U_n,\cdot\rangle$ are groups. Multiplication of complex numbers is associative and both $U$ and $U_n$ contain 1,which is an identity for multiplication.For $e^{i\theta} \in U$,the computation

\begin{align} e^{i\theta}\cdot e^{i(2\pi-\theta)}=e^{2\pi i}=1 \end{align}

shows that every element of $U_n$ has an inverse. Thus $\langle U,\cdot\rangle$ and $\langle U_n,\cdot\rangle$ are groups.Because $\langle \mathbb{R}_c,+_c\rangle$ is isomorphic to $\langle U,\cdot\rangle$,we see that $\langle \mathbb{R}_c,+_c\rangle$ is a group for all $c\in\mathbb{R}^+$.Similarly,the fact that $\langle \mathbb{Z}_n,+_n\rangle$ is isomorphic to $\langle U,\cdot\rangle$ shows that $\langle \mathbb{Z}_n,+_n\rangle$ is a group for all $n\in \mathbb{Z}^+$.


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