Solving the Mystery of the Table: Explaining $g = (123)$

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Discussion Overview

The discussion revolves around the computation of permutations represented in cycle notation, specifically focusing on the permutation $g = (123)$ and its application to various inputs, including the identity function and another permutation $(12)$. The scope includes mathematical reasoning and technical explanations related to group theory and permutations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants explain that $(123)$ maps $1$ to $2$, $2$ to $3$, and $3$ to $1$, while $(1)$ represents the identity function.
  • Participants detail the computation of $(123)(1)$, concluding that it results in $(123)$, as each element is mapped according to the defined cycles.
  • There is a repeated explanation of how to compute $(123)(12)$, with participants noting that $1$ is mapped to $3$, $2$ is mapped to $2$, and $3$ is mapped to $1$, resulting in $(13)$.
  • Multiple posts reiterate the same computations and mappings, suggesting a focus on clarity and understanding of the process rather than introducing new information.

Areas of Agreement / Disagreement

Participants appear to agree on the mappings defined by the permutations and the results of the computations. However, there is no explicit discussion of alternative interpretations or methods, leaving the conversation primarily focused on the established mappings without contestation.

Contextual Notes

The discussion does not address any potential limitations or assumptions underlying the computations, nor does it explore the broader implications of the results in group theory.

Who May Find This Useful

This discussion may be useful for individuals interested in understanding permutations in group theory, particularly those seeking clarification on cycle notation and function composition.

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Could someone please explain how they're getting the answers in the table, for example $g = (123)$.
 

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$$(123)(1)$$

$(123)$ represents the function that maps $1$ to $2$, $2$ to $3$ and $3$ to $1$.
$(1)$ represents the identity function, i.e. the function that maps $1$ to $1$, $2$ to $2$ and $3$ to $3$.

So to compute $(123)(1)$ we do the following:

From the right cycle we have that $1$ is mapped to $1$, and at the first cycle $1$ is mapped to $2$, therefore we get that $1$ is mapped to $2$.
From the right cycle we have that $2$ is mapped to $2$, and at the first cycle $2$ is mapped to $3$, therefore we get that $2$ is mapped to $3$.
From the right cycle we have that $3$ is mapped to $3$, and at the first cycle $3$ is mapped to $1$, therefore we get that $3$ is mapped to $1$.

So, we get $(123)(1)=(123)$.
$$(123)(12)$$

$(123)$ represents the function that maps $1$ to $2$, $2$ to $3$ and $3$ to $1$.
$(12)$ represents the function that maps $1$ to $2$, $2$ to $1$ and $3$ to $3$.

So to compute $(123)(12)$ we do the following:

From the right cycle we have that $1$ is mapped to $2$, and at the first cycle $2$ is mapped to $3$, therefore we get that $1$ is mapped to $3$.
From the right cycle we have that $2$ is mapped to $1$, and at the first cycle $1$ is mapped to $2$, therefore we get that $2$ is mapped to $2$.
From the right cycle we have that $3$ is mapped to $3$, and at the first cycle $3$ is mapped to $1$, therefore we get that $3$ is mapped to $1$.

So, we get $(123)(12)=(13)$.
 
mathmari said:
$$(123)(1)$$

$(123)$ represents the function that maps $1$ to $2$, $2$ to $3$ and $3$ to $1$.
$(1)$ represents the identity function, i.e. the function that maps $1$ to $1$, $2$ to $2$ and $3$ to $3$.

So to compute $(123)(1)$ we do the following:

From the right cycle we have that $1$ is mapped to $1$, and at the first cycle $1$ is mapped to $2$, therefore we get that $1$ is mapped to $2$.
From the right cycle we have that $2$ is mapped to $2$, and at the first cycle $2$ is mapped to $3$, therefore we get that $2$ is mapped to $3$.
From the right cycle we have that $3$ is mapped to $3$, and at the first cycle $3$ is mapped to $1$, therefore we get that $3$ is mapped to $1$.

So, we get $(123)(1)=(123)$.
$$(123)(12)$$

$(123)$ represents the function that maps $1$ to $2$, $2$ to $3$ and $3$ to $1$.
$(12)$ represents the function that maps $1$ to $2$, $2$ to $1$ and $3$ to $3$.

So to compute $(123)(12)$ we do the following:

From the right cycle we have that $1$ is mapped to $2$, and at the first cycle $2$ is mapped to $3$, therefore we get that $1$ is mapped to $3$.
From the right cycle we have that $2$ is mapped to $1$, and at the first cycle $1$ is mapped to $2$, therefore we get that $2$ is mapped to $2$.
From the right cycle we have that $3$ is mapped to $3$, and at the first cycle $3$ is mapped to $1$, therefore we get that $3$ is mapped to $1$.

So, we get $(123)(12)=(13)$.
Wonderful explanations, thanks!
 
mathmari said:
$$(123)(1)$$

$(123)$ represents the function that maps $1$ to $2$, $2$ to $3$ and $3$ to $1$.
$(1)$ represents the identity function, i.e. the function that maps $1$ to $1$, $2$ to $2$ and $3$ to $3$.

So to compute $(123)(1)$ we do the following:

From the right cycle we have that $1$ is mapped to $1$, and at the first cycle $1$ is mapped to $2$, therefore we get that $1$ is mapped to $2$.
From the right cycle we have that $2$ is mapped to $2$, and at the first cycle $2$ is mapped to $3$, therefore we get that $2$ is mapped to $3$.
From the right cycle we have that $3$ is mapped to $3$, and at the first cycle $3$ is mapped to $1$, therefore we get that $3$ is mapped to $1$.

So, we get $(123)(1)=(123)$.

$$(123)(12)$$

$(123)$ represents the function that maps $1$ to $2$, $2$ to $3$ and $3$ to $1$.
$(12)$ represents the function that maps $1$ to $2$, $2$ to $1$ and $3$ to $3$.

So to compute $(123)(12)$ we do the following:

From the right cycle we have that $1$ is mapped to $2$, and at the first cycle $2$ is mapped to $3$, therefore we get that $1$ is mapped to $3$.
From the right cycle we have that $2$ is mapped to $1$, and at the first cycle $1$ is mapped to $2$, therefore we get that $2$ is mapped to $2$.
From the right cycle we have that $3$ is mapped to $3$, and at the first cycle $3$ is mapped to $1$, therefore we get that $3$ is mapped to $1$.

So, we get $(123)(12)=(13)$.

Spoken like a pro.
 

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