What is the meaning of $mathbb{Z}_{5}^{+}$ (and $mathbb{Z}_{5}^{*}$) in group theory?












3












$begingroup$


What does $mathbb{Z}_{5}^{+}$ mean? I know $mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(mathbb{Z}_{5},+)$. I'm confused as to the meaning of this notation?



Edit:



Furthermore, what is the meaning of $mathbb{Z}_{5}^{*}$. The context I saw this in seems to imply this is a group but if it was the same thing as $(mathbb{Z}_{5},*)$ it couldn't be since $(mathbb{Z}_{5},*)$ lacks some inverses.






group-theory finite-groups notation modular-arithmetic







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$endgroup$












  • $begingroup$
    The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
    $endgroup$
    – M. Nestor
    Dec 19 '18 at 3:41


















3












$begingroup$


What does $mathbb{Z}_{5}^{+}$ mean? I know $mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(mathbb{Z}_{5},+)$. I'm confused as to the meaning of this notation?



Edit:



Furthermore, what is the meaning of $mathbb{Z}_{5}^{*}$. The context I saw this in seems to imply this is a group but if it was the same thing as $(mathbb{Z}_{5},*)$ it couldn't be since $(mathbb{Z}_{5},*)$ lacks some inverses.






group-theory finite-groups notation modular-arithmetic







share|cite|improve this question











$endgroup$












  • $begingroup$
    The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
    $endgroup$
    – M. Nestor
    Dec 19 '18 at 3:41
















3












3








3





$begingroup$


What does $mathbb{Z}_{5}^{+}$ mean? I know $mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(mathbb{Z}_{5},+)$. I'm confused as to the meaning of this notation?



Edit:



Furthermore, what is the meaning of $mathbb{Z}_{5}^{*}$. The context I saw this in seems to imply this is a group but if it was the same thing as $(mathbb{Z}_{5},*)$ it couldn't be since $(mathbb{Z}_{5},*)$ lacks some inverses.






group-theory finite-groups notation modular-arithmetic







share|cite|improve this question











$endgroup$




What does $mathbb{Z}_{5}^{+}$ mean? I know $mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(mathbb{Z}_{5},+)$. I'm confused as to the meaning of this notation?



Edit:



Furthermore, what is the meaning of $mathbb{Z}_{5}^{*}$. The context I saw this in seems to imply this is a group but if it was the same thing as $(mathbb{Z}_{5},*)$ it couldn't be since $(mathbb{Z}_{5},*)$ lacks some inverses.






group-theory finite-groups notation modular-arithmetic




group-theory finite-groups notation modular-arithmetic






share|cite|improve this question















share|cite|improve this question













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edited Dec 21 '18 at 17:11









Shaun

9,869113684




9,869113684










asked Dec 19 '18 at 2:21









PolymorphismPrincePolymorphismPrince

1948




1948












  • $begingroup$
    The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
    $endgroup$
    – M. Nestor
    Dec 19 '18 at 3:41




















  • $begingroup$
    The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
    $endgroup$
    – M. Nestor
    Dec 19 '18 at 3:41


















$begingroup$
The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
$endgroup$
– M. Nestor
Dec 19 '18 at 3:41






$begingroup$
The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
$endgroup$
– M. Nestor
Dec 19 '18 at 3:41












2 Answers
2






active

oldest

votes


















3












$begingroup$

For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.



On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.



For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.






share|cite|improve this answer









$endgroup$





















    3












    $begingroup$

    The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.



    [For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
      $endgroup$
      – PolymorphismPrince
      Dec 19 '18 at 2:43










    • $begingroup$
      More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
      $endgroup$
      – Aaron
      Dec 19 '18 at 3:27












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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.



    On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.



    For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.






    share|cite|improve this answer









    $endgroup$


















      3












      $begingroup$

      For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.



      On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.



      For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.






      share|cite|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.



        On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.



        For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.






        share|cite|improve this answer









        $endgroup$



        For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.



        On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.



        For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 19 '18 at 5:35









        Chris CusterChris Custer

        14.3k3827




        14.3k3827























            3












            $begingroup$

            The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.



            [For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
              $endgroup$
              – PolymorphismPrince
              Dec 19 '18 at 2:43










            • $begingroup$
              More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
              $endgroup$
              – Aaron
              Dec 19 '18 at 3:27
















            3












            $begingroup$

            The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.



            [For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
              $endgroup$
              – PolymorphismPrince
              Dec 19 '18 at 2:43










            • $begingroup$
              More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
              $endgroup$
              – Aaron
              Dec 19 '18 at 3:27














            3












            3








            3





            $begingroup$

            The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.



            [For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]






            share|cite|improve this answer









            $endgroup$



            The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.



            [For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 19 '18 at 2:39









            Clive NewsteadClive Newstead

            52k474136




            52k474136












            • $begingroup$
              Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
              $endgroup$
              – PolymorphismPrince
              Dec 19 '18 at 2:43










            • $begingroup$
              More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
              $endgroup$
              – Aaron
              Dec 19 '18 at 3:27


















            • $begingroup$
              Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
              $endgroup$
              – PolymorphismPrince
              Dec 19 '18 at 2:43










            • $begingroup$
              More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
              $endgroup$
              – Aaron
              Dec 19 '18 at 3:27
















            $begingroup$
            Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
            $endgroup$
            – PolymorphismPrince
            Dec 19 '18 at 2:43




            $begingroup$
            Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
            $endgroup$
            – PolymorphismPrince
            Dec 19 '18 at 2:43












            $begingroup$
            More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
            $endgroup$
            – Aaron
            Dec 19 '18 at 3:27




            $begingroup$
            More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
            $endgroup$
            – Aaron
            Dec 19 '18 at 3:27


















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