Tour:Factsheet one (beginners)

Here are some facts we have seen so far:


 * 1) A group can be defined as a set with three operations: a binary operation, called the multiplication or product, an inverse map, and an identity element. These three operations are subject to the conditions of associativity, identity element and inverses.
 * 2) A group can also be defined as a set with an associative binary operation, called the multiplication or product, such that there exists an identity element and every element has an inverse.
 * 3) The two definitions of group are equivalent because the identity element is unique, and inverses, when they exist, are unique. We'll see proofs of this in part two.
 * 4) Subgroups are subsets that are closed under all the group operations. Just having a subset that is closed under the multiplication operation is not enough: for instance, the nonzero integers are closed under multiplication inside the group of nonzero rationals under multiplication, but they don't form a subgroup.
 * 5) There are two somewhat different ways of defining subgroups: one, where we demand closure under all the group operations, and the other, where we demand closure only under the multiplication, but insist that the subset form a group with this induced operation. The two definitions are equivalent, and we'll see a proof of this in part two.
 * 6) There is a third way of defining subgroups in terms of left quotients of elements. We'll see, in part two, a proof of the equivalence of this with the other definitions.
 * 7) A very special class of groups is the abelian groups. A group is abelian iff any two elements commute i.e. the binary operation on it is commutative. Any subgroup of an abelian group is abelian.
 * 8) The multiplication operation in an abelian group is often denoted additively.
 * 9) The trivial group is the group with just one element.

Notational conventions

 * 1) Groups are often denoted by letters like $$G,H,K$$
 * 2) The multiplication operation is denoted by $$*$$ or $$\cdot$$, or by omission. Because of associativity of multiplication, we can omit parentheses when multiplying more than two elements, and, when the context is clear, omit the multiplication symbol as well.
 * 3) The inverse operation is denoted by a superscript of $$-1$$. The superscript applies only to the immediately preceding expression. Thus $$xy^{-1}$$ is $$x * (y^{-1})$$ and not $$(x * y)^{-1}$$
 * 4) A product of the same element with itself many times is denoted by a power of that element. So $$x^n = xxx\ldots$$ $$n$$ times
 * 5) The multiplicative identity is denoted by $$e$$ or $$1$$
 * 6) For abelian groups, $$+$$ denotes the addition, and iterated sum is denoted by integer multiplication. So $$nx = x + x + \ldots + x$$ done $$n$$ times.
 * 7) $$-$$ denotes the additive inverse in an abelian group, and $$0$$ denotes the additive identity.
 * 8) Subgroups in general are denoted by the $$\le$$ sign. So $$H \le G$$ means $$H$$ is a subgroup of $$G$$. We can also say $$H \subseteq G$$, but the latter is also used for mere subsets, that aren't subgroups.