Theorems

 

Theorem :

Let 𝐻,𝐾 be cyclic groups of order 𝑚, 𝑛 respectively, where 𝑚, 𝑛 are relatively 

prime. Then 𝐻 × 𝐾 is a cyclic group of order 𝑚𝑛.

Theorem: 

Let 𝐺 = 𝐻 × 𝐾 and 𝐻1 be a normal subgroup of 𝐻. Then 𝐻1 is normal in 𝐺.

Theorem: 

Let 𝐺 = 𝐻 × 𝐾. Then the factor group 𝐺/𝐾≅ 𝐻.

Theorem:

 let 𝐺 be a group. Then

a) the derived group 𝐺′is normal subgroup of 𝐺,

b) the factor group 𝐺/𝐺′is abelian,

c) if 𝐾 is a normal subgroup of 𝐺 such that 𝐺/𝐾 is abelian then 𝐺′ ⊆ 𝐾.

Theorem:

 Let 𝜁 (𝐺) be the centre and 𝐼(𝐺) be the inner automorphism of a group 𝐺. Then 𝐺

𝜁( 𝐺 )≅ 𝐼(𝐺).

Theorem:

The set 𝐼(𝐺) of all inner automorphism of a group 𝐺 is a normal subgroup of 

𝐴(𝐺).

Theorem:

 Let 𝐺 be a group. The mapping 𝜑 ∶ 𝐺 ⟶ 𝐺 defined by𝜑 (𝑔 )= 𝑔−1, 𝑔 ∈ 𝐺

is an automorphism if and only if 𝐺 is abelian.

Theorem :

The set 𝐴 𝐺 of all automorphism of 𝐺 form a group.

Theorem:

 Let 𝐺 be a direct product of its two normal subgroups 𝐻,𝐾 with 𝐻 ∩ 𝐾 = {𝑒} and 

𝐺 = 𝐻𝐾. Then

i. Each element of 𝐻 is permutable with every element of 𝐾. i.e, h𝑘 = 𝑘h , for all h ∈ 𝐻, 𝑘 ∈ 𝐾

ii. Every element of 𝐺 is uniquely expressible as 𝑔 = h𝑘, for all h∈ 𝐻, 𝑘 ∈ 𝐾

iii. 𝐺 ≅ 𝐻 × 𝐾.

Conjugation as an Automorphism:

Let 𝐺 be a group, 𝑎 ∈ 𝐺. Define a mapping 𝐼𝑎 ∶ 𝐺 ⟶ 𝐺 by 

𝐼𝑎 (𝑔 )= 𝑎𝑔𝑎−1, for all 𝑔 ∈ 𝐺.

Then 𝐼𝑎 is an automorphism.

Embedding: 

An embedding of a group 𝐺 into a group 𝐺′is simply a monomorphism of 𝐺 into 𝐺′. in other words, if 𝐺 is embedded in a group 𝐺′then 𝐺′contains a subgroup 𝐻′ isomorphic to G.

Simple group :

 A group is said to be simple group if it has no proper normal subgroups.

Complete Group: 

if the centre 𝜁( G) of a group 𝐺 is trevial and very automorphism of 𝐺 is an inner 

automorphism, 𝐺 is called a complete group.

Derived Group OR Commutator Subgroup:

 Let 𝐺 be a group and 𝐺′ be a subgroup of 𝐺. Then 𝐺′is said to be a commutator subgroup, if it is generated by a set of commutators of 𝐺.