Let H be a subgroup of G. If for every a G, we have aHa^{-1} = H, then H is called a *normal subgroup* of G.
Let R be a ring. An *ideal* of R is a subring I of R such that if a I, r R, then a · r I and r · a I.
If H is a subgroup of G then aH = {a · h | a in G, h in H} is a *left coset* Ha = {h · a | a in G, h in H} is a *right coset*
Let G_{1} and G_{2} be groups. An *isomorphism* from G_{1} to G_{2} is an injection :G_{1}G_{2} such that
(a · b) = (a) · (b) (a,b G_{1}).
A function *f* : AB is said to be *injective* (or *one to one*) if whenever *f*(x) = *f*(y), we have x = y.
A function that is both one-to-one and onto, that is if f(x) = f(y) then x = y, and for every y of the domain there is an x of the range so that f(x) = y.
A function *f* : AB is said to be *surjective* (or *onto*) if for every y B there exists x A such that *f*(x) = y.
Let G and H be groups. A function f :GH which satisfies f(g · g') = f(g) · f(g') for all g, g' G is called a *homomorphism* of G into H.
If f:GH is a homomorphism. The *kernel* of f is ker(f) = {x G | f(x) = 1_{H}}.
Let G_{1} and G_{2} be groups. An *isomorphism* from G_{1} to G_{2} is an injection :G_{1}G_{2} such that
(a · b) = (a) · (b) (a,b G_{1} if is also surjective G_{1} is *isomorphic* to G_{2}).
Let H be a subgroup of G. If for every a G, we have aHa^{-1} = H, then H is called a *normal subgroup* of G.
The set of integers.
commutative
The set of rational numbers.
The set of real numbers.
The set of natural numbers.
The complex numbers.
The set residue classes mod n.
The set of reduced residue classes mod n.
For all a,b,c G, we have
a·(b·c) = (a·b)·c.
There exists an element *e* of G which has the property
*e* · x = x · *e* = x
for all x G.
For every x G, the exists *y* G such that
x · *y* = *y* · x = *e*.
A commutative ring R is an *integral domain* if R contains no zero divisors. In other words, R is an integral domain if the product of any two nonzero elements of R is nonzero.
Let R and S be rings. A *ring homomorphism* from R to S is a function *f* :RS such that for a,b R, we have (a) *f*(a + b) = *f*(a) + *f*(b),, (b) *f*(a · b) = *f*(a) · *f*(b). If <*f* is injective, *f* is a *ring isomorphism*
A group is considered *cyclic* if it is generated by a single element.
:n(n) = the number of integers a (1 __<__ a __<__ n) such that (a,n) = 1.
If E is a field, then a *subfield of E* is a subset of E which is also a field with respect to the operations of E.
Let R be a ring with identity 1, and let a R. If a has an inverse with respect to multiplication, then we say that a is a *unit* of R.
We say that R is a *unique factorization domain* if the following two conditions are satisfied: (1) If x R^{x} is not a unit of R, then x can be written as a product of irreducible elements of R.and (2) If x R^{x} is not a unit of R, and if x = _{1} ... _{s} = _{1} ... _{t} are two expressions of x as a product of irreducible elements, then s = t and it is possible to renumber _{1} ... _{s} so that _{1} and _{1} are associates (1 __<__ i __<__ s).
A ring R having the property that every ideal is principal is called a *principal ideal ring* (PIR). If, in addition, R is an integral domain, then R is called a *principal ideal domain* (PID).
Let R be a ring x,y R. We say that x and y are *associates* if x = y for some unit of R.
If f has a leading coefficient 1, then we say that f is monic.
We say that is *algebraic over F* if there exists a nonzero polynomial *f* F[X] such that *f*() = 0. If is not algebraic over F, then we say that is *transcendental over F*.
Let V be a vector space over F. A *basis of V* is a subset {**e**_{i}} of V (finite or infinite) with the property that every element **v** of V can be uniquely written in the form
**v** = _{i}**e**_{i}.
An ideal M of R is said to be a *maximal ideal* if (a) M is a proper ideal and (b) if A is a proper ideal containing M, then M = A.
Let V be a vector space. If V has a finite set of generators, then we say that V is *finite-dimensional* and we define its *dimension*, denoted dim_{F}V, to be the minimum possible number of elements in such a set. If V does not have a finite set of generators, then we say the V is *infinite-dimensional*, and we define dim_{F}V to be
Let E be algebraic over F. the zero of a monic irreducible polynomial p F[X]. The polynomial p is called the irreducible polynomial of over F denoted Irr_{F}(,X)
The subfield F(_{1},...,_{n}) of E gotten by adjoining all zeros of *f*(X) to F is called the *splitting field* of *f*.
An ideal A of R is said to be *proper* if AR.
Let F be a field. A *normal extension* of F is an extension of F obtained by adjoining all zeros of a finite set of polynomials {*f*_{1},...,*f*_{m}}, *f*_{j} F[X].
An *F-automorphism* of E is an F-isomorphism of E onto itself. The set of all F-automorphisms of E will be denoted Gal(E/F). Gal(E/F) is called the *Galois group of the extension E over F*
If *f* : GH is a homomorphism, then G/ker(*f*) im(*f*)
A group G of order n is said to be *solvable* if it has a composition series whose composition factors are cyclic of prime order.