1) a,b in H implies that ab in H

2) a in H implies that a^(-1) is in H.

1. If H is a subgroup of G, a in G, then Ha = __

2. Ha is called a__

(1) {ha|h in H}

2) right coset of H in G

the number of distinct right cosets in G

o(G)/o(H)

a^Φ(n) ≡ 1 (modn)

Euler

o(HK) = [o(H)*o(K)]/o(H∩K)

If Φ is a homomorphism of G into G', then

1) Φ(e) =

2)Φ(x^(-1))=

1) e'

2)Φ(x)^(-1)

Cauchy's for Abelian

Suppose G is a finite abelian group and p | o(G), where p prime. Then

Sylow's for Abelian

If G is an abelian group of order o(G), and if p is a prime s.t. p^a | o(G), P^(a+1) † o(G), then

Sylow Corollary

If G abelian of order o(G) and p^a | o(G), p^(a+1) † o(G), then 

Let Φ be a homomorphism of G onto G' with kernal K. For H' a subgroup of G' let H be defined by H= {x in G | Φ(x) in H'}. Then

1) H is

2) if H' is normal in G', then 

3)This association

1) a subgroup of G and H contains K

2) H is normal in G

3) sets up a one-to-one mapping from the set of all subgroups of G' onto the set of all subgroups of G which contain K.

T_g:G->G by xT_g = g^(-1)xg for x in G.

(Cayley)

Every group is isomorphic

If G is a group, H a subgroup of G, and S is the set of all right cosets of H in G, then

1) there is

2) and the kernal

1) a homorphism Θ of G into A(S)

2) of θ is the largest normal subgroup of G which is contained in H.

If G is a finite group, then c_a = (1)

2) in other words, the number of elements conjugate to a

1) o(G)/o(N(A))

2) is the index of the normalizer of a in G.

a in Z iff N(a) = 1)

2) if G is finite, a in Z iff

1) G

2) o(N(a)) = o(G)

Cauchy

If p is a prime and p | o(G), then

Sylow

If p is a prime number and p^a|o(G), then

A non-empty set R is said to be a ring if for all a,b in R:

1) (R,+) is 

2) R is

3) multiplication

1) an abelian group with identity 0 and additive inverse -a.

2) closed and associative under • (both directions)

3) distributes over addition (both directions)

A mapping φ: R-> R' is said to be a homomorphism of R if, for all a,b in R:

1) φ(a+b)

2) φ(ab)

1) = φ(a) + φ(b)

2) = φ(a)φ(b)

If φ is a homomorphism of R into R', then for every a in R:

1) φ(0) =

2) φ(-a) =

1) 0

2) -φ(a)

A non-empty subset U of R is said to be a (two-sided) ideal of R if:

1) (U,+) is

2) For every u in U and r in R,

1) a subset of (R,+)

2) both ur and ru are in U

Let φ: R->R' surjective with kernal U, and ideal of R. Then

1) R' is isomorphic to

2) There is a 1-1 correspondence between

3) This correspondence is achieved by associating with an ideal I' in R'

4) With I so defined, R/I is

1) R/U

2) the set of ideal of R' and the set of ideals of R which contain U

3) the ideal I in R defined by I = {x in R| φ(x) in W'}

4) isomorphic to R'/I'

1) A ring R can be embedded in R' uf there is

2) If R and R' have unit elements, then

3) R' will be called an over-ring or extension of R if

1) an isomorphism of R into R'.

2) this isomorphism takes 1 onto 1'.

3) R can be embedded in R'.

(Gauss' Lemma)

If the primitive polynomial f(x) can be factored as the product of two polynomials have rational coefficients, it can be factored as the

Eisenstein

Let f(x)=a0+a1x+...+anxⁿ be a polynomial with INTEGER coefficients. Suppose that for some prime number p, p†an, p|a1, p|a2,..., p|a0, p² †a0. Then f(x) is