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Subgroup

$Subgroup$

H

is a subgroup of

(G, \cdot)

:

(H, \cdot)

is a group

$trivial subgroup$

Trivial subgroup of

G

:

{e_{G}}

$Corollary$

e_{G} = e_{H}

a_{G}^{- 1} = a_{H}^{- 1}

$Corollary$

H_{1}, H_{2}

is a subgroup of

G

\Rightarrow

H_{1} \cap H_{2}

is a subgroup of

G

$set dot$

ST = {s \cdot t ∣ s \in S, t \in T}

S^{- 1} = {s^{- 1} ∣ s \in S}

$Element dot set$

a S = {a \cdot s ∣ s \in S} S a = {s \cdot a ∣ s \in S} a S b = {a \cdot s \cdot b ∣ s \in S, t \in T}

$Number of element$

∣ a S ∣ = ∣ S b ∣ = ∣ a S b ∣ = ∣ S ∣

$Corollary$

1. H is a subgroup of G 2.\forall a, b \in H (ab \in H \land a^{- 1} \in H) 3.\forall a, b \in H (a^{- 1} b \in H)

$Corollary$

1. H is a subgroup of G 2 H \subset H \land H^{- 1} \subset H 3^{'} H^{- 1} H \subset H

$Generated subgroup$

⟨ S ⟩

is subgroup generated from

S

:

1. S \subset ⟨ S ⟩ 2. H is a subgroup of (G, \cdot) \Rightarrow ∣ ⟨ S ⟩ ∣ \leq ∣ H ∣

$Centralizer$

Centralizer of

S

Z_{G} (S) := {x \in G ∣\forall s \in S (x \cdot s \cdot x^{- 1} = s)}

$Normalizer$

Normalizer of

S

N_{G} (S) := {x \in G ∣ x S x^{- 1} = S}

$Corollary$

S = {a} \Rightarrow Z_{G} (S) = N_{G} (S)

$Center$

Center of

G

:

Z_{G} (G)

$Regular subgroup$

N

is regular subgroup of

G

:

\forall x \in G (x N x^{- 1} \subset N)

G ▹ N

$Corollary$

1. G ▹ H 2 N (H) = G

$Corollary$

1. N (H) ▹ H 2. G ▹ Z (G)

$Simple group$

G

is a simple group :

G ▹ N \Rightarrow (N = {e_{G}} \lor N = G)