26. Groups. Theorems on uniqueness of the unit and the inverse element; order of an element.

A non-empty set G is called a group if a binary algebraic operation is given in G (often called multiplication), i.e. a unique element is determined for every ordered pair (a, b) of elements of G (their product) and the following conditions hold:

1. for alla, b, c  G (the multiplication is associative).

2. There is a unit e  G such that for alla  G (existence of unit).

3. For any a  G there is an inverse element such that(existence of inverse element).

A group G is called commutative or abelian if for alla, b  G. In an abelian group the binary operation is sometimes called addition and therefore the product is replaced by the sum and is denoted by a + b. In this case a unit is denoted by zero 0, and an inverse for a element is called opposite is denoted by – a.

Theorem 1. There is the only unit in a group.

Proof. Let be a group. Assume the contrary: there are at least two unitsandof.

Then since is a unit of the group, we havefor alla  G.

Consequently, . On other hand, sinceis a unit of the group, we also have. Thus,.

Theorem 2. There is the only inverse element for every element of a group.

Proof. Let be a group. Assume the contrary: there exists an elementhaving at least two inverse elementsandfor, i.e.and. Then we have:

. 

The number of elements of a group G (if it is finite) is called the order of group G is denoted by |G|. In this case G is called finite. If the set G is infinite then the group G is called infinite.

27. Groups: subgroups, left and right cosets, factor-groups.

Example 2. Let Z₃ be the set consisting of numbers 0, 1 and 2, and let a binary operation be addition of numbers by module 3. Does the set Z₃ form a group? If yes, is it cyclic? If this set is a cyclic group, determine a generating element.

Solution: The addition by module 3 is determined as follows:

We see that for all a,b of Z₃ we have a + b Z₃.

The addition by module 3 is associative and commutative. The number 0 is the zero-element of Z₃. 1 is inverse for 2, and 2 is inverse for 1. Thus, Z₃ is a group. The number 1 is a generating element of Z₃, i.e. Z₃ is a cyclic group.

A subset H of a group G is called a subgroup of group G if H is group with respect to the operation given in G. If is a subgroup of, we denote this by. Ifand, we denote this by.

For example the set H = {1} from Example 1 is a subgroup of the group G.

Example of subgroups.

1) The set of integers with the operation of addition is a subgroup of the group.

2) The set of positive rational numbers with the operation of multiplication is a subgroup of the group.

A subgroup H of a group G is called normal in G if for any elements h  H, g  G the element also belongs toH. An element of group of kind is calledconjugate with an element h by g. If is a normal subgroup of, we denote this by.

Example. is a normal subgroup of.

Two groups G₁ and G₂ (with operations  and  respectively) are called isomorphic if there such one-to-one correspondence (bijection) mapping that for any two elementsa and b of G the following equality holds:

The notation of isomorphism of groups: .

Let H be a subgroup of G. A left coset of an element g  G on the subgroup H is called the set

A right coset is determined analogously.

The group G is partitioned into pairwise non-intersecting left (right) cosets on the subgroup H, and the cardinality of any coset is equal to the cardinality of H.

Theorem of Lagrange. The order of a finite group is divided on the order of any its subgroup.

Proof. Let be a group, andis a subgroup of. Let, i.e.is the order of, and. Obviously,. Consider all left cosets ofon:. We have:for all,for every, and. Then obviously, i.e.is divided on.

In general . If a subgroupH is normal in G then for allg  G. And in this case the set G/H of cosets of group G on subgroup H is a group with respect to the operation of multiplication of cosets defined by the following equality:

This group is called the factor group of group G on normal subgroup H.

Example. Consider – the group of integers with the operation of addition. Letbe the set of integers that are multiple to 5, i.e. the following set:. Then obviouslywith the operation of addition is a subgroup of.

Consider the left cosets on the subgroup :where. Thenis partitioned into pairwise non-intersecting left cosetsand.