599: Group as Direct Sum of Finite Number of Normal Subgroups Is Group as Direct Sum of Any Reordered and Combined Normal Subgroups|T.B.P.

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description/proof of that group as direct sum of finite number of normal subgroups is group as direct sum of any reordered and combined normal subgroups

Topics

About: group

Starting Context

The reader knows a definition of group as direct sum of finite number of normal subgroups.
The reader admits the proposition that for any group, the product of any finite number of subgroups is associative.
The reader admits the proposition that for any group, the product of any finite number of normal subgroups is commutative and is a normal subgroup.

Target Context

The reader will have a description and a proof of the proposition that any group as direct sum of finite number of normal subgroups is the group as direct sum of any reordered and combined normal subgroups.

Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description

Here is the rules of Structured Description.

Entities:

G

\in {the groups}

{G_{1}, . . ., G_{n}}

\subseteq {the normal subgroups of G}

{G_{σ_{1}}, . . ., G_{σ_{n}}}

σ \in {the permutations of (1, . . ., n)}

{G_{σ_{1}} . . . G_{σ_{l_{1}}}, . . ., G_{σ_{l_{1} + . . . + l_{k - 1} + 1}} . . . G_{σ_{n}}}

n = l_{1} + . . . + l_{k}

//

Statements:

G = the direct sum of {G_{1}, . . ., G_{n}}

⟹

G = the direct sum of {{G_{σ_{1}} . . . G_{σ_{l_{1}}}, . . ., G_{σ_{l_{1} + . . . + l_{k - 1} + 1}} . . . G_{σ_{n}}}

.
//

2: Natural Language Description

For any group,

G

, any normal subgroups,

G_{1}, . . ., G_{n}

, of

G

such that

G

is the direct sum of

{G_{1}, . . ., G_{n}}

{G_{σ_{1}}, . . ., G_{σ_{n}}}

, where

σ

is any permutation of

(1, . . ., n)

, and any

{G_{σ_{1}} . . . G_{σ_{l_{1}}}, . . ., G_{σ_{l_{1} + . . . + l_{k - 1} + 1}} . . . G_{σ_{n}}}

, where

n = l_{1} + . . . + l_{k}

G

is the direct sum of

{G_{σ_{1}} . . . G_{σ_{l_{1}}}, . . ., G_{σ_{l_{1} + . . . + l_{k - 1} + 1}} . . . G_{σ_{n}}}

3: Proof

Let us see that

G

is the direct sum of

{G_{σ_{1}}, . . ., G_{σ_{n}}}

.

For each

G_{σ_{j}} \in {G_{σ_{1}}, . . ., G_{σ_{n}}}

σ_{j} = k \in {1, . . ., n}

, and

G_{σ_{j}} \cap (G_{σ_{1}} . . . G_{σ_{j - 1}} \hat{G_{σ_{j}}} G_{σ_{j + 1}} . . . G_{σ_{n}}) = G_{k} \cap (G_{1} . . . G_{k - 1} \hat{G_{k}} G_{k + 1} . . . G_{n}) = {1}

, by the proposition that for any group, the product of any finite number of normal subgroups is commutative and is a normal subgroup.

G = G_{1} . . . G_{n} = G_{σ_{1}} . . . G_{σ_{n}}

, by the proposition that for any group, the product of any finite number of normal subgroups is commutative and is a normal subgroup.

For the simplicity of notations, hereafter, let us think of

{G_{1} . . . G_{l_{1}}, . . ., G_{l_{1} + . . . + l_{k - 1} + 1} . . . G_{n}}

instead of

{G_{σ_{1}} . . . G_{σ_{l_{1}}}, . . ., G_{σ_{l_{1} + . . . + l_{k - 1} + 1}} . . . G_{σ_{n}}}

, because we already know that the permutation does not cause any problem.

Let us see that

G

is the direct sum of

{G_{1} . . . G_{l_{1}}, . . ., G_{l_{1} + . . . + l_{k - 1} + 1} . . . G_{n}}

.

Let us see that for each

G_{l_{1} + . . . + l_{j - 1} + 1} . . . G_{l_{1} + . . . + l_{j - 1} + l_{j}}

S := G_{l_{1} + . . . + l_{j - 1} + 1} . . . G_{l_{1} + . . . + l_{j - 1} + l_{j}} \cap ((G_{1} . . . G_{l_{1}}) . . . (G_{l_{1} + . . . + l_{j - 2} + 1} . . . G_{l_{1} + . . . + l_{j - 2} + l_{j - 1}}) \hat{(G_{l_{1} + . . . + l_{j - 1} + 1} . . . G_{l_{1} + . . . + l_{j - 1} + l_{j}})} (G_{l_{1} + . . . + l_{j} + 1} . . . G_{l_{1} + . . . + l_{j} + l_{j + 1}}) . . . (G_{l_{1} + . . . + l_{k - 1} + 1} . . . G_{n})) = {1}

.

For each

p \in S

p = p_{l_{1} + . . . + l_{j - 1} + 1} . . . p_{l_{1} + . . . + l_{j - 1} + l_{j}} = (p_{1} . . . p_{l_{1}}) . . . (p_{l_{1} + . . . + l_{j - 2} + 1} . . . p_{l_{1} + . . . + l_{j - 2} + l_{j - 1}}) \hat{(p_{l_{1} + . . . + l_{j - 1} + 1} . . . p_{l_{1} + . . . + l_{j - 1} + l_{j}})} (p_{l_{1} + . . . + l_{j} + 1} . . . p_{l_{1} + . . . + l_{j} + l_{j + 1}}) . . . (p_{l_{1} + . . . + l_{k - 1} + 1} . . . p_{n})

, where

p_{j} \in G_{j}

p_{l_{1} + . . . + l_{j - 1} + 1} = (p_{1} . . . p_{l_{1}}) . . . (p_{l_{1} + . . . + l_{j - 2} + 1} . . . p_{l_{1} + . . . + l_{j - 2} + l_{j - 1}}) \hat{(p_{l_{1} + . . . + l_{j - 1} + 1} . . . p_{l_{1} + . . . + l_{j - 1} + l_{j}})} (p_{l_{1} + . . . + l_{j} + 1} . . . p_{l_{1} + . . . + l_{j} + l_{j + 1}}) . . . (p_{l_{1} + . . . + l_{k - 1} + 1} . . . p_{n}) ({p_{l_{1} + . . . + l_{j - 1} + l_{j}}}^{- 1} . . . {p_{l_{1} + . . . + l_{j - 1} + 2}}^{- 1}) \in G_{l_{1} + . . . + l_{j - 1} + 1} \cap ((G_{1} . . . G_{l_{1}}) . . . (G_{l_{1} + . . . + l_{j - 2} + 1} . . . G_{l_{1} + . . . + l_{j - 2} + l_{j - 1}}) \hat{(G_{l_{1} + . . . + l_{j - 1} + 1} . . . G_{l_{1} + . . . + l_{j - 1} + l_{j}})} (G_{l_{1} + . . . + l_{j} + 1} . . . G_{l_{1} + . . . + l_{j} + l_{j + 1}}) . . . (G_{l_{1} + . . . + l_{k - 1} + 1} . . . G_{n}) (G_{l_{1} + . . . + l_{j - 1} + l_{j}} . . . G_{l_{1} + . . . + l_{j - 1} + 2}) \subseteq G_{l_{1} + . . . + l_{j - 1} + 1} \cap G_{1} . . . G_{l_{1} + . . . + l_{j - 1}} \hat{G_{l_{1} + . . . + l_{j - 1} + 1}} G_{l_{1} + . . . + l_{j - 1} + 2} . . . G_{n}) = {1}

, which implies that

p_{l_{1} + . . . + l_{j - 1} + 1} = (p_{1} . . . p_{l_{1}}) . . . (p_{l_{1} + . . . + l_{j - 2} + 1} . . . p_{l_{1} + . . . + l_{j - 2} + l_{j - 1}}) \hat{(p_{l_{1} + . . . + l_{j - 1} + 1} . . . p_{l_{1} + . . . + l_{j - 1} + l_{j}})} (p_{l_{1} + . . . + l_{j} + 1} . . . p_{l_{1} + . . . + l_{j} + l_{j + 1}}) . . . (p_{l_{1} + . . . + l_{k - 1} + 1} . . . p_{n}) ({p_{l_{1} + . . . + l_{j - 1} + l_{j}}}^{- 1} . . . {p_{l_{1} + . . . + l_{j - 1} + 2}}^{- 1}) = 1

p_{l_{1} + . . . + l_{j - 1} + 2} = (p_{1} . . . p_{l_{1}}) . . . (p_{l_{1} + . . . + l_{j - 2} + 1} . . . p_{l_{1} + . . . + l_{j - 2} + l_{j - 1}}) \hat{(p_{l_{1} + . . . + l_{j - 1} + 1} . . . p_{l_{1} + . . . + l_{j - 1} + l_{j}})} (p_{l_{1} + . . . + l_{j} + 1} . . . p_{l_{1} + . . . + l_{j} + l_{j + 1}}) . . . (p_{l_{1} + . . . + l_{k - 1} + 1} . . . p_{n}) ({p_{l_{1} + . . . + l_{j - 1} + l_{j}}}^{- 1} . . . {p_{l_{1} + . . . + l_{j - 1} + 3}}^{- 1}) \in G_{l_{1} + . . . + l_{j - 1} + 2} \cap ((G_{1} . . . G_{l_{1}}) . . . (G_{l_{1} + . . . + l_{j - 2} + 1} . . . G_{l_{1} + . . . + l_{j - 2} + l_{j - 1}}) \hat{(G_{l_{1} + . . . + l_{j - 1} + 1} . . . G_{l_{1} + . . . + l_{j - 1} + l_{j}})} (G_{l_{1} + . . . + l_{j} + 1} . . . G_{l_{1} + . . . + l_{j} + l_{j + 1}}) . . . (G_{l_{1} + . . . + l_{k - 1} + 1} . . . G_{n}) (G_{l_{1} + . . . + l_{j - 1} + l_{j}} . . . G_{l_{1} + . . . + l_{j - 1} + 3})) \subseteq G_{l_{1} + . . . + l_{j - 1} + 2} \cap (G_{1} . . . G_{l_{1} + . . . + l_{j - 1} + 1} \hat{G_{l_{1} + . . . + l_{j - 1} + 2}} G_{l_{1} + . . . + l_{j - 1} + 3} . . . G_{n}) = {1}

, which implies that

p_{l_{1} + . . . + l_{j - 1} + 2} = (p_{1} . . . p_{l_{1}}) . . . (p_{l_{1} + . . . + l_{j - 2} + 1} . . . p_{l_{1} + . . . + l_{j - 2} + l_{j - 1}}) \hat{(p_{l_{1} + . . . + l_{j - 1} + 1} . . . p_{l_{1} + . . . + l_{j - 1} + l_{j}})} (p_{l_{1} + . . . + l_{j} + 1} . . . p_{l_{1} + . . . + l_{j} + l_{j + 1}}) . . . (p_{l_{1} + . . . + l_{k - 1} + 1} . . . p_{n}) ({p_{l_{1} + . . . + l_{j - 1} + l_{j}}}^{- 1} . . . {p_{l_{1} + . . . + l_{j - 1} + 3}}^{- 1}) = 1

.

And so on. After all,

p_{l_{1} + . . . + l_{j - 1} + 1} = . . . = p_{l_{1} + . . . + l_{j - 1} + l_{j}} = 1

, so,

p = p_{l_{1} + . . . + l_{j - 1} + 1} . . . p_{l_{1} + . . . + l_{j - 1} + l_{j}} = 1

. So,

S = {1}

G = G_{1} . . . G_{n} = (G_{1} . . . G_{l_{1}}) . . . (G_{l_{1} + . . . + l_{k - 1} + 1} . . . G_{n})

, by the proposition that for any group, the product of any finite number of subgroups is associative.

References

The Bias Planet
Definitions and Propositions
School Mathematics from Higher Viewpoints
To Disentangle Confusing Terms or Discourses
Information Tables
Exploiting LibreOffice or OpenOffice
Exposing the Java Programming Language
Exposing C++
Exposing Python
Exposing C#
Exposing Gradle
Exposing Git
UNO Dispatch Commands

2024-06-30

599: Group as Direct Sum of Finite Number of Normal Subgroups Is Group as Direct Sum of Any Reordered and Combined Normal Subgroups

Topics

The table of contents of this article

Starting Context

Target Context

Orientation

Main Body

1: Structured Description

2: Natural Language Description

3: Proof

References