Tour:Factsheet one (beginners)

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)
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Below is a list of some important facts that can be proved directly from the definitions we have seen so far of group, subgroup, trivial group, identity element, Abelian group. We'll see these facts (and more) with full proof, in part two of the guided tour. Links to the full proof statements are provided. To make best use of the tour as a learning experience, it is suggested that you try to prove these statements yourself.

1. The identity element in a group is unique. This comes from a more general fact: in any magma (i.e. a set with binary operation), if there exists a left neutral element (a left identity for the binary operation) and a right neutral element, they are both equal. For full proof, refer: Neutral element#Any left and right neutral element are equal
2. Now that we know what the trivial group is, we can interpret this in a somewhat more sophisticated way. For any group, there is a unique subgroup of it that looks like the trivial group. That subgroup is the one and only identity element of the group.
3. In particular, any group is nonempty.
4. Given any element, it has a unique inverse. The proof of this uses associativity. The proof works in a slightly greater generality: if we have a monoid (a set with associative binary operation and a neutral element i.e. a multiplicative identity) and an element happens to have a left inverse and also happens to have a right inverse, the left and right inverse must be equal. For full proof, refer: Inverse element#Equality of left and right inverses
5. The upshot: the binary operation of the group determines the other two operations.
6. 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.
7. However, if we have a nonempty subset that is closed under the operation $(x,y)\mapsto x{y}^{-1}$ it is a subgroup. For full proof, refer: Sufficiency of subgroup condition
8. 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.
9. The multiplication operation in an Abelian group is denoted additively.

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.
3. The inverse operation is denoted by a superscript of $-1$. The superscript applies only to the immediately preceding expression. Thus $x{y}^{-1}$ 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 $x^n=xxx\dots$ $n$ 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 $nx=x+x+\dots +x$ done $n$ times.
7. $-$ denotes the additive inverse in an Abelian group
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.