Intro to abstract algebra lecture notes 1 PDF

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Professor Shalane - Intro to abstract algebra lecture notes 1...

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Order of an element Smallest number of times so that g^n=ggg...=e

Index of a subgroup H in a group G The number of cosets of H in G (G:H)

Partition If G acts on A, orbits partition A, or their union is A, and conjugacy classes partition G

Permutation An arrangement of elements of an ordered list S into a 1-1 correspondence with S itself

Symmetric Groups Sn. The group of all permutations of n symbols

Alternating Group An. The group of all even permutations in Sn. Contains all products of an even number of transpositions

Even Permutation A permutation that can be expressed by an even number of transpositions

Automorphism An isomorphism from a group to itself

Simple Group A group with no proper, non-trivial normal subgroup

Normal Subgroup A subgroup of G in which gH=Hg for all g in G

Quotient Group A group obtained by aggregating similar elements of a larger group using an equivalence relation. Elements are the fibers of the homomorphism with the group operation defined

Characteristic Subgroup A subgroup that is mapped to itself by every automorphism of the group

Stabilizer Elements in a group G which fix an element in a group A {ga0=a0}

Orbit Oa. {ga0 for all g in G}. A subgroup of A. Partition A

Sylow p-Subgroup A subgroup of order p^n with p prime and p^n the highest power of p that divides the order of G

Direct Product of Groups An operation that takes 2 groups G and H and constructs a new group GxH. Forms the Cartesian product of ordered pairs

Conjugacy Class The orbits of G acting on itself by conjugation. {gag^-1 for all g in G}. Partition G

Centralizer All elements of G that commute with a. CG(a). {all g in G : ga=ag}

Ring Homomorphism Given 2 rings R, S, there is a ring homomorphism from R to S if $(a+b)=$(a)+$(b) and $(ab)=$(a)$(b)

Ideal A subring of R in which IR is contained in I and RI is contained in I

Kernel All elements of G that commute with every a in G

Integral Domain A commutative ring with 1 and no zero divisors

Field A commutative ring with an identity and every non-zero element has a multiplicative inverse

Zero Divisor Nonzero elements in a ring such that ab=0

Unit An element in a ring that has the identity element and a multiplicative inverse on both sides (uv=1, vu=1)

Diamond Isomorphism Theorem If N is a normal subgroup of G and H is a subgroup of G then H/H intersect N is isomorphic to HN/N and H intersect N is a normal subgroup of H and HN is a subgroup of G

Lattice Isomorphism Theorem If N is a normal subgroup of G then the lattice diagram of subgroups of G/N corresponds to the lattice diagram of subgroups of G that contain N

Cayley's Theorem If G is a finite group of order n, the G is isomorphic to a subgroup of Sn

Cauchy's Theorem If G is a finite group and p divides the order of G with p prime, then G contains an element of order p

Lagrange's Theorem

If H is a subgroup of a finite group G, then the order of H is a factor of the order of G

Class Equation The order of G is equal to the order of the center of G plus the sum of the orders of the conjugacy classes

Sylow Theorem Part 1 If p^r divides the order of G then G has a subgroup of order p^r

Sylow Theorem Part 2 If p^n is the highest power of p that divides the order of G, then a subgroup of order p^n is a Sylow pSubgroup of G. All Sylow p-subgroups of G are conjugate to each other

Sylow Theorem Part 3 The number of different sylow p-subgroups is written np and np is a divisor of the order of G and np is congruent to 1 modulo p

Fundamental Theorem of Finite Abelian Groups A finite abelian group is isomorphic to a direct product of (several) cyclic groups

Recognizing Direct Products Theorem If K and N are normal subgroups of a group G and K intersect N is {1G}, then KxN is isomorphic to KN which is a normal subgroup of G

Fermat's Little Theorem If p is prime and p doesn't divide a, then a^p-1 is congruent to 1 modulo p

Index of Stabilizer Theorem The size of an orbit is the index of the stabilizer (order of Oa=(G:Ga))

Fundamental Isomorphism Theorem for Rings

If R,S are rings and there exists a ring homomorphism from R to S, then R/ker($) is isomorphic to $(R) which is contained in S...