Tour:Factsheet one (beginners)

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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. We'll see the proof of this in part two.
2. The trivial group is the group with just one element. In every group, we can find a copy of the trivial group -- as the subset with just the identity element.
3. In particular, any group is nonempty, and the smallest group is the trivial group.
4. Given any element, it has a unique inverse. The proof of this uses associativity. We'll see this proof in part two.
5. The binary operation of a 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: for instance, the nonzero integers are closed under multiplication inside the group of nonzero rationals under multiplication, but they don't form a subgroup.
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.

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.
This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour). If you found anything difficult or unclear, make a note of it; it is likely to be resolved by the end of the tour.
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