What does the notation $(G,.)$ mean in group theory? I have seen in places that $.$ implies the binary operation multiplication on group $G$. But then, why do we show an abelian group as $(G, +)$? And what is additive and multiplicative notation?
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$\begingroup$$(G,\cdot)$ denotes the ordered pair with first entry the underlying set $G$ of the group and second entry the law of composition $\cdot$ of the group. The ordered pair notation is very common in other parts of mathematics, for instance for metric spaces $(M,d)$ or topological spaces $(X,\tau)$ and so on (the general pattern is $(\mathrm{set},\mathrm{structure \ on \ the \ set})$; it is useful because one does not have to define the notion of equality for groups, metric spaces and topological spaces and so on separately as it can be shown that two ordered pairs $(x,y)$ and $(x',y')$ are equal as sets iff $x=x'$ and $y=y'$. Hence two groups are equal iff their underlying set and their law of composition are equal. It is however very common to denote the group $(G,\cdot)$ simply by $G$ (similarily for metric and topological spaces) if the law of composition (metric, topology) is understood. Also a group is by definition an ordered pair $(G,\cdot)$ with $G$ a set and $\cdot$ a law of composition on $G$ subject to the group axioms (some people say that a group is a set $G$ "together with a law of composition on $G$" by which they simply mean that $(G,\cdot)$ is an ordered pair).
Additive notation refers to denoting the law of composition by $+$ (multiplicative notation $\cdot$), the unit element by $0$ (multiplicative notation $1$) and the inverse of $x$ by $-x$ (multiplicative notation $x^{-1}$). As others have pointed out, this is by convention often done when the law of composition is commutative.
$\endgroup$ $\begingroup$ "But is there any particular underlying reason? – Artemisia"I can't reply yet above, but I would think that (G,+) is used when the law of composition is commutative because (for example) matrix ADDITION is commutative, but matrix MULTIPLICATION is not, although both operations on matrices are associative. This convention may have arisen here. Recall that it is probably unnecessary to explain WHY conventions are the way they are in most cases....
$\endgroup$ $\begingroup$the notation $(G,.)$ mean you have a group where the operation is called "." If you write $(G,+)$, the name of the operation is $+$. the $.$ and $+$ are just name for operations.
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