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Thread: Cycle notation

  1. MHB Craftsman

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    #1


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

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  3. MHB Master
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    #2
    $$(123)(1)$$

    $(123)$ 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 $(123)(12)=(13)$.

  4. MHB Craftsman

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    #3 Thread Author
    Quote Originally Posted by mathmari View Post
    $$(123)(1)$$

    $(123)$ 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!

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    #4
    Quote Originally Posted by mathmari View Post
    $$(123)(1)$$

    $(123)$ 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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