There exists operator
a order of group is a number of elements in a group.
A subgroup of is normal if its left coset and right coset coincide, that is if for all .
Let be a normal subgroup of . then the cosets of form a group and is the factor group.
A group is simple if it is nontrivial and has no proper nontrivial normal subgroups.
Let be a group and let . then the subgroup this group is the cyclic group of generated by . and the order of this cyclic subgroup called the order of (element) a.
A subgroup of is a conjugate subgroup oof if for some .
Let is a prime. Let be a finite group and let p divide . then has an element of order and consequently a subgroup of order .
For is a subgroup of , divides .
Let be a prime. A group is a p-group if every element in has order a power of the prime .
Sylow p-subgroup is a maximal p-sbugroup of .
Let is a finite group and where and where does not divide then
Let and be sylow p-subgroups. and are conjugate subgroup of .
If divide , the number of sylow p-subgroups is and divide .
A ring is a nonempty set with two operations such that
Let be a ring. there exists a least positive number such that , for . is said to have characteristic n. if there not exits in , this called characteristc zero.
A ring which has communtative multiplication is Communtative Ring.
and if contains multiplicative unity , is said to be Ring with unity(identity).
A nonzero element in a ring is said to be a zero divisor if there exists nonzero such that .
an element in is said to be invertible or unit if there exsits such that .
is a communtative ring with unity and no zero divisor is called a Integral domain.
the ring is an integral domain.
with nonzero unity in which every nonzero element is a unit is called a Division ring.
A field is a communtative division ring.
Let be a subring of . satisfying the properties
is an ideal.
the cyclic group is an ideal in
If an extension of a is of finite dimension as vector space over , then is a finite extension of degree over . let \[E:F\] be the degree of over .
Let be a finite extension of a field . the number of isomorphism of onto subfield of a closure leaving fixed is the index of over .
A extension field of is the splitting field over of the polynomial if splits in , that is, if with , and .
is said to be a seperable extension of if every elements of is the simple root such that for and .