661: 6-Elements Group Cannot Have 2 3-Elements Subgroups That Share Only Identity

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description/proof of that 6-elements group cannot have 2 3-elements subgroups that share only identity

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Topics

About: group

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Proof

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Starting Context

• The reader knows a definition of group.

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Target Context

• The reader will have a description and a proof of the proposition that any 6-elements group cannot have any 2 3-elements subgroups that share only the identity.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$G$: $\in \{\text{ the groups }\}$
$G_1$: $\in \{\text{ the subgroups of }G\}$
$G_2$: $\in \{\text{ the subgroups of }G\}$
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Statements:
(
$|G|=6$
$\wedge$
$|G_1|=3$
$\wedge$
$G_1\cap G_2=\{1\}$
)
$⟹$
$|G_2|\ne 3$
//

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2: Natural Language Description

Any 6-elements group, $G$, cannot have any 2 3-elements subgroups, $G_1,G_2$, such that $G_1\cap G_2=\{1\}$.

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3: Proof

Let us suppose that there were $G_1=\{1,a_1,a_2\}$ and $G_2=\{1,b_1,b_2\}$ such that $G_1\cap G_2=\{1\}$.

$\{1,a_1,a_2,b_1,b_2\}$ would be distinct, and there would be the leftover, $c\in G$.

What would be $b_1{a}_1$? It could not be $1$, because $b_1{a}_1=1$ would imply $b_1={a_1}^{-1}\in G_1\cap G_2=\{1\}$, which would imply $b_1=1$, a contradiction. It could not be $a_j$, because $b_1{a}_1=a_j$ would imply $b_1=a_j{a_1}^{-1}\in G_1\cap G_2=\{1\}$, which would imply $b_1=1$, a contradiction. It could not be $b_j$, because $b_1{a}_1=b_j$ would imply $a_1={b_1}^{-1}{b}_j\in G_1\cap G_2=\{1\}$, which would imply $a_1=1$, a contradiction. So, $b_1{a}_1=c$. But likewise, $b_1{a}_2=c$, then, $a_1={b_1}^{-1}c=a_2$, a contradiction. So, $b_1{a}_1$ cannot have any valid value.

So, the supposition is false.

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References

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