Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.

Fundamentals of Group Theory

Basic concepts and theorems concerning groups.

Sequence of Expressions

Definition

Group

Beginner

A group is a set

G

with an operation

\cdot

such that:\n1.

(ab)c = a(bc)

\n2.

\exists e \in G

s.t.

ae = ea = a

\n3.

\forall a \in G, \exists b \in G

s.t.

ab = ba = e

Definition

Abelian Group

Beginner

A group

(G, \cdot)

is abelian if for all

a, b \in G

a \cdot b = b \cdot a

Theorem

Uniqueness of Identity

Beginner

e

and

e'

are identity elements of a group

G

, then

e = e'

Theorem

Uniqueness of Inverse

Beginner

For every

a \in G

, there exists a unique

b \in G

such that

ab = ba = e

Definition

Subgroup

Beginner

A subset

H \subseteq G

is a subgroup if

H

is a group under the operation of

G

restricted to

H

Theorem

Subgroup Test

Intermediate

A non-empty subset

H

of a group

G

is a subgroup if and only if for all

a, b \in H

ab^{-1} \in H

Theorem

Lagrange's Theorem

Intermediate

G

is a finite group and

H

is a subgroup of

G

, then the order (number of elements) of

H

divides the order of

G

.\n

|H| \text{ divides } |G|

Definition

Coset

Intermediate

For a subgroup

H \subseteq G

and

g \in G

, the left coset is

gH = \{gh : h \in H\}

and the right coset is

Hg = \{hg : h \in H\}

Definition

Normal Subgroup

Intermediate

A subgroup

N

G

is normal (denoted

N \trianglelefteq G

) if

gng^{-1} \in N

for all

n \in N

and

g \in G

Theorem

Quotient Group Construction

Advanced

N \trianglelefteq G

, the quotient group

G/N

is the set of all cosets of

N

G

with the operation

(aN)(bN) = (ab)N

Theorem

First Isomorphism Theorem

Advanced

Let

\phi: G \to H

be a group homomorphism. Then

G / \ker(\phi) \cong \text{im}(\phi)