Abstract Algebra Midterm

Front 1

binary operation

Back 1

it is a function on A such that A x A ->A

Front 2

closed subset

Back 2

let H be a subset of A. a subset is closed under a binary operation if a * b is in H for all a,b in H.

Front 3

commutative law

Back 3

a*b=b*a

Front 4

associative law

Back 4

(a*b)*c=a*(b*c)

Front 5

identity element

Back 5

an element e in A is an identity element for * if a*e=a=e*a.

Front 6

inverse

Back 6

if a is in A, an inverse for a is an element a' in A such that a*a;=e and a'*a=e

Front 7

isomorphism

Back 7

let (G,*) and (H,~) be groups. f:G->H. f is an isomorphism of the binary operations if f is bijetive and if f(a*b)=f(a)~f(b) for all a,b in G.

Front 8

group

Back 8

the pair (G,*) is a group if it satisfies associative identity and inverses law

Front 9

abelian group

Back 9

if it also satisifes commutative law

Front 10

order

Back 10

lGl is the cardinality of the group=how many elements

Front 11

a =bmod n

Back 11

a-b=kn

Front 12

properties of isomorphic groups

Back 12

f:G->H, j:H->K


f(eG)=eH.
if a in aG, then f(a')=[f(a)]' where the first is in G and teh second is in H.
identity map 1G:G->G is an isomorphism
function f^-1 is an isomorphism
function j of f is an isomorphism

Front 13

properties of groups

Back 13

1.e'=e
2. if ac=bc then a=b
3. if ca=cb, then a=b
4. (a')'=a
5. (ab)'=b'a'
6. if ba=e, then b=a'
7. if ab=e, then b=a'

Front 14

left identity element

Back 14

e*a=a for a in A

Front 15

subgroup

Back 15

let H be a subset of G and G is a group. H is a subgroup of G if
1. H is closed under *
2. (H,*) is a group

Front 16

properties of subgroup

Back 16

if H is a subgrp of G
1. the identity element of G is in H, and it is the identity element of H
2. the inverse op in H is the same as teh inverse op in G
3. if G is cyclic, then H is cyclic

Front 17

H is a subgroup of G if

Back 17

1. e is in H
2. if a,b are in H, then a*b is in H
3. if a is in H, then a' is in H
are all true
OR
a. H is not an empty set
b. if a, b in H, then a*b is in H
c. if a is in H, then a' is in H

Front 18

cyclic group

Back 18

if G=<a> for some a in G, the element a is a generator of G. if cyclic, then abelian

Front 19

cyclic subgroup

Back 19

let H be a subgrp of G. H is a cyclic subgrp of G if H=<a> for some a in G. a is a generator of H

Front 20

infinite order

Back 20

if <a> is finite, the order of a (lal) is the cardinality of <a>. if <a> is infinite, then a has infinite order

Front 21

properties of cyclic groups

Back 21

G is cyclic
if G is infinite, G is isomorph to Z
if lGl=n for some n in N, then G is isomorph to Zn

Front 22

permutation

Back 22

a permutation of A is bijective map A->A. set of all perms of A is S_A.

Front 23

relations

Back 23

let sigma be in S_A. let ~ be the relation on A defined by a~b if and only if b=sigma^n(a) for n in Z for all a,b in A

Front 24

orbit

Back 24

let A be a set and let sigma be in S_A. equivalence classes of ~ are orbits of sigma.

Front 25

cycle

Back 25

the permutation sigma is a cyucle if it has at most one orbit with more than one element. the length of a cycles is the number of elements in teh largest orbit.

Front 26

transposition

Back 26

cycle of length 2

Front 27

transpositions properties

Back 27

let n be in N and let sigma be in S_n. suppose n is greater htna or equal to 2. then all representations of sigma as a product of transpositions have an even or odd number

Front 28

even/odd transpositions

Back 28

the perm sigma is even or odd if it is the product of an even number or an odd number of transpositions

Front 29

A_n

Back 29

even permutations, subgroup of Sn.
lAnl=n!/2

Front 30

product binary operation

Back 30

on H x K, binary op defined by (h1.k1)(h2,k2)=(h1h2,k1k2) for all (h1,k1),(h2,k2) in H x K

Front 31

cosets

Back 31

left coset: equivalence class of a with respect to ~L is aH. let H be a subgrp of G and a is in G. the left coset of a is teh set aH.
right coset: opposite.

aH=bH if an only if a^-1b is in H
Ha=Hb if an only if ab^-1 is in H

Front 32

index

Back 32

the index of H in G denoted (G:H) is the number of left cosets of G with respect to H

Front 33

normal subgroup

Back 33

the following are equivalent, and H is a normal subgroup of G if any of them are true
a. gHg^-1 is a subset of H for all g in G
b. gHg^-1=H for all g in G
c. gH=Hg for all g in G

Front 34

quotient group

Back 34

Let H be a normal subgroup of G. the set G/H with binary op given by (aH)(bH)=(ab)H for all a,b, in G is quotient group of G by H

Front 35

simple group

Back 35

if it has no nontrivial proper normal subgroups