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Finite Groups

Groups with a finite number of elements, fundamental in algebra and combinatorics.

Sequence of Expressions

Theorem

Includes Lagrange's Theorem, Sylow Theorems, Cayley's Theorem, and the Classification of Finite Simple Groups.

Theorem

H

is a subgroup of a finite group

G

, then

|G| = [G:H] \cdot |H|

, where

[G:H]

is the index of

H

G

Theorem

For a prime

p

dividing

|G|

, Sylow

p

-subgroups exist, form a single conjugacy class, and their number

n_p

satisfies

n_p \equiv 1 \pmod p

and

n_p \mid |G|

Theorem

Every group

G

is isomorphic to a subgroup of the symmetric group acting on

G

\text{Sym}(G)

Theorem

G

is a finite group of order

p^a q^b

where

p

and

q

are primes, then

G

is solvable.

Theorem

Every finite group of odd order is solvable.

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|