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Group.
A group is a pair $(G,\,*)$, where $G$ is a nonempty set and “$*$”
is a binary operation on $G$, such that the following conditions hold:

For any $a,b,c\in G$, $(a*b)*c=a*(b*c)$. (Associativity of the operation).

There is an element $e\in G$ such that $g*e=e*g=g$ for any $g\in G$. (Existence of identity element).

For any $g\in G$ there exists an element $h$ such that $g*h=h*g=e$. (Existence of inverses).
If $G$ is a group under *, then * is referred to as the group operation of $G$.
Usually, the symbol “$*$” is omitted and we write $ab$ for $a*b$. Sometimes, the symbol “$+$” is used to represent the operation, especially when the group is abelian.
It can be proved that there is only one identity element, and that for every element there is only one inverse. Because of this we usually denote the inverse of $a$ as $a^{{1}}$ or $a$ when we are using additive notation. The identity element is also called neutral element due to its behavior with respect to the operation, and thus $a^{{1}}$ is sometimes (although uncommonly) called the neutralizing element of $a$. An element of a group besides the identity element is sometimes called a nontrivial element.
Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions. In fact, much of the study of groups themselves is conducted using group actions.
Mathematics Subject Classification
14F99 no label found08A99 no label found20A05 no label found2000 no label found83C99 no label found32C05 no label found Forums
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Attached Articles
uniqueness of inverse (for groups) by waj
a characterization of groups by yark
groups of small order by Daume
inverse of a product by pahio
alternative definition of group by pahio
division in group by pahio
inverse of inverse in a group by cvalente
word by juanman
redundancy of twosidedness in definition of group by pahio
Corrections
neutral and neutralizin element by pahio ✓
nontrivial element by yark ✓
grammar by Wkbj79 ✓
group operation by Wkbj79 ✓
Comments
I+ll write more later
I dont have more time now...
I+ll add examples and stuff later
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