The group is a notion even more primitive than the ring, because in case of a group we will be only considering one operation (depending on the context it will be denoted by or ). So a set with a fixed neutral element and with operation is a group if
- for any , ,
- for any , ,
- for any there exists (denoted by ) such that .
If additionally the operation is commutative, i.e. for any , , then such a group is called an abelian group. Actually, in this case is usually used to denote the operation (thus we use instead of , and instead of then).
Notice that if is a commutative ring with an identity, it is an abelian group with its operation (and neutral element ) as well, which is called the additive group of this ring. Also the set of units of a ring , denoted by is an abelian group with operation and neutral element . This group is called the multiplicative group of .
The set of all matrices over a field with operation of multiplication and neutral element , denoted by , is a group. Notice that this group is not abelian.
A subset is a subgroup if and for also and . We denote this by .
E.g. the set of all matrices over with determinant is a subgroup of . It is denoted by . Thus, .
A mapping , where are groups, is a homomorphism, if for any , . It follows that and .
Notice also that and the kernel .
E.g. , where is modulo is a homomorphism between the additive groups of those rings. Also, is yet another example of a homomorphism.
Order of an element and of a group
The order of a group is its cardinality is it is finite. In the other case we say that the order of the group is infinite.
The order of an element , , is the least natural number , such that , or , is such does not exist.
E.g. is the additive group of , since (zero is the neutral element, and is the operation here).
A group is a torsion group if all its elements are of finite order.
Cosets of a subgroup
If , and , then is a left coset of . Notice that if and only if .
The set of all left cosets is denoted by . Its cardinality is called the index of in and is denoted by . Lagrange’s Theorem states that in every finite group , with , we have .
A subgroup is normal, if for any , the set . This is denoted by .
It is important to notice that for every homomorphism its kernel is a normal subgroup.
Generators, cyclic groups
We say that a subgroup is generated by a set , if is the least subgroup containing , which is denoted by . In particular, then (if ),
If , then is the set of generators of the group . If , we simply write and then the group is called a cyclic group.
Notice that all subgroups of a cyclic group are cyclic, and if an order of the group is finite, there is exactly one subgroup of order equal to a divisor of the order of the whole group.
Groups of permutations
A permutation of a given set is a bijection from this set onto itself, which can be also understood as an ordering of its elements in the case of finite sets. The set of all permutations of a set is denoted by . Permutations can be composed, so along with operation it is a group.
In particular, we will take interest in permutations of the set . The set of such permutations will be denoted by . E.g. the function , , is an element of . We will denote the permutations as two rows of numbers, in the upper row the arguments and in the lower the values corresponding to them. Thus the considered above permutation can be denoted as:
Thus, we have for example:
A permutation of a set is a cycle if there are such that , and is constant on all other elements. Such a permutation will be denoted simply as . Thus, in the above example, .
We can easily notice that any permutation can be decomposed into an composition of disjoint cycles, e.g.:
Notice also that the order of each cycle is equal to its length, so the order of each permutation is the leach common multiple of the length of cycles in its decomposition into disjoint cycles.
Finally, each cycle can be decomposed into (not disjoint) transpositions (cycles of length ). Indeed, .
One can prove, that although there are multiple possibilities of decomposition of a permutation into transpositions, for a given permutation all the decomposition have either even or odd number of transpositions. Thus, a permutation is even if it can be written as a composition of an even number of transpositions, and otherwise it is odd. The set of all even permutations is a normal subgroup of all permutations , which can be checked quite easily.
Cayley Theorem states that every group is isomorphic to a subgroup of the group . Indeed, every element can be identified with a permutation of the group of the form .
If is a normal subgroup, then the set of posets can be regarded also as a group with operation and . This makes sense only if is a normal subgroup, because only then we can be sure that the operation is well-defined, by which we mean that if and , then .
There is a natural homomorphism, which is onto, defined as . Obviously, .
Moreover, the homomorhpism theorem states that if is a homomorphism, then there exists exactly one monomorphism , such that , where is the projection defined above. In particular, the groups and are isomorphic.
Direct products and sums
Given groups (finitely or infinitely many, ), the Cartesian product of these groups is the Catesian product equipped by the coordinatewise operation. Its subgroup is the so called direct product of all elements with finitely many non-neutral coordinates.
For groups with notation we use the term direct sum and write .
On can notice that if is a group and is a family of groups with , and , and also for any , then . In such case we also say that is the inner product (or inner sum in the case of groups with ) of its subgroups , i.e. .
It is worth noticing that , where is a factorization of . Actually, it is true for any finite abelian group in the following sense, , where and . Finally, every abelian group of order , where is a prime number can be described as a direct sum of some of its cyclic subgroups. This means that every finite abelian group is a direct sum of some of its cyclic subgroups.
Centre, commutator, conjugate elements
The set called the centre of the group is a normal subgroup of , which “measures” how much is commutative. On the other hand let , and is a commutator of . It is also a normal subgroup of and it measures how is non-commutative. In particular, a homomorphic image of is abelian if and only if the kernel contains .
The element is called an adjunct of and the set of all adjuncts of will be denoted by .
Notice that , where , which means that in the case of finite groups the is a divisor of , and obviously since is a union of classes of adjuncts their cardinalities sum up to the order of .
Notice also that . Even more can be said in the case of permutations. Namely, are conjuncts if and only if their decompositions into disjoint cycles contain the same number of cycles of the same lenght. E.g. and are conjuncts, because for
we get .
We say that a group acts on a set , if there exists a homomorphism . Usually we write instead of . Cayley’s Theorem describes an action of a group on itself. Another obvious example is the action of on the set of verteces of -polygon.
For , the set is called an orbit of . Obviously orbits give a partition of . The set is the set of fixed points. Notice that for every , . Finally is a subgroup of , and is the group of stabilizers of .
It is crucial to notice that . In particular this means that the cardinality of every orbit is a divisor of the order of . This gives as a sum of divisors of .
An important collorary follows from consideration of group actions, which is Cauchy Theorem. This theorem states that if is a finite group and a prime number is such that , then there exists such that .