Abstract Algebra Chapter 7

Front 1

Left coset of H in G containing a, denoted aH
Right coset of H in G containing a, denoted Ha

Coset representative

Back 1

Given a subset H of a group G and an element a of G,

aH={ah: h belongs H}
Ha={ha : h belongs H}

a is coset representative.

Front 2

Properties of Cosets

Back 2

If H ≤ G and a and b belong to G, then

(1) a belongs aH
(2) aH = H if and only if a belong H
(3) aH = bH or aH ∩ bH = Ø
(4) aH = bH if and only if a^-1b belongs H
(5) |aH| = |bH|
(6) aH = Ha if and only if H = aHa-1
(7) aH ≤ G if and only if a belongs H

Front 3

Lagrange’s Theorem

Back 3

If G is a finite group and H ≤ G, then |H| divides |G|. Furthermore, the number of distinct left or right cosets of H in G is |G|/|H|.

Front 4

Index of a subgroup, denoted |G:H|

Back 4

Given H ≤ G, the number of distinct left cosets of H in G.

Front 5

Corollaries of Lagrange’s Theorem

Back 5

(1) If G is a finite group and H ≤ G, then |G:H| =|G|/|H|.
(2) For each element a of G, |a| divides |G|.
(3) Groups of prime order are cyclic.
(4) If G is a finite group, then for any element a of G, a^|G| = e.
(5) Fermat’s Little Theorem

Front 6

Fermat’s Little Theorem

Back 6

For every integer a and every prime p, a^p mod p = a mod p.

Front 7

Classification of groups of order 2p

Back 7

Let G be a group of order 2p, where p is a prime greater than 2. Then G is isomorphic either to Z sub 2p or to D sub of p.

Front 8

Stabilizer of i in G, denoted stab subs of G (i)

Back 8

Let G be a group of permutations of a set S. Then
stabG(i) = { @ belong G ; @(i)= i} .

Front 9

Orbit of s in G, denoted orbG(s)

Back 9

Let G be a group of permutations of a set S. Then
orbG(s) = { @(s) ; @ belongs G} .

Front 10

Orbit-stabilizer theorem

Back 10

Let G be a finite group of permutations of a set S. Then, for any i in S, |G| = |orbG(i)| |stabG(i)|