Order of Group and LCM of group elements. [closed]












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Is order of Group equal to lowest common multiples of order of group elements.?If yes, what are the conditions?










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closed as off-topic by Shailesh, Chinnapparaj R, user10354138, Brahadeesh, KReiser Nov 26 at 8:35


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    Is order of Group equal to lowest common multiples of order of group elements.?If yes, what are the conditions?










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    closed as off-topic by Shailesh, Chinnapparaj R, user10354138, Brahadeesh, KReiser Nov 26 at 8:35


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shailesh, Chinnapparaj R, user10354138, Brahadeesh, KReiser

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      Is order of Group equal to lowest common multiples of order of group elements.?If yes, what are the conditions?










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      Is order of Group equal to lowest common multiples of order of group elements.?If yes, what are the conditions?







      group-theory






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      asked Nov 26 at 3:22









      ViperX_2

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      closed as off-topic by Shailesh, Chinnapparaj R, user10354138, Brahadeesh, KReiser Nov 26 at 8:35


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shailesh, Chinnapparaj R, user10354138, Brahadeesh, KReiser

      If this question can be reworded to fit the rules in the help center, please edit the question.




      closed as off-topic by Shailesh, Chinnapparaj R, user10354138, Brahadeesh, KReiser Nov 26 at 8:35


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Shailesh, Chinnapparaj R, user10354138, Brahadeesh, KReiser

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          No, for a finite abelian group the $rm lcm$ of the orders is the expt (not order) of the group, namely



          $begin{eqnarray}rm{bf Proposition}quad maxord(G) !&,=,&rm expt(G) text{ for a finite abelian group} G, i.e.\ \
          rm max { ord(g) : : g in G} !&=&rm min { n>0 : : g^n = 1 forall g in G}end{eqnarray}$



          Proof $ $ By the lemma below, $rm: S, =, { ord(g) : :g in G }$ is a finite
          set of naturals closed under$rm lcm$.



          Hence every $rm s in S:$ is a
          divisor of the max elt $rm: m $ [else $rm: lcm(s,m) > m,$],$ $ so $rm m = expt(G)$.



          Lemma $ $ A finite abelian group $rm:G:$ has an lcm-closed order set, i.e. with $rm: o(X) = $ order of $rm: X$



          $$rm X,Y in G Rightarrow exists Z in G: o(Z) = lcm(o(X),o(Y))$$



          Proof $ $ By induction on $rm: o(X), o(Y). $ If it's $:1:$ then trivially $rm:Z = 1$. $ $ Otherwise



          write $rm o(X) =: AP,: o(Y) = BP', P'|,P = p^m > 1, $ prime $rm: p:$ coprime to $rm: A,B.$



          Then $rm: o(X^P) = A, o(Y^{P'}) = B. $ By induction there's a $rm: Z:$ with $rm : o(Z) = lcm(A,B)$



          so $rm o(X^A: Z): =: P lcm(A,B): =: lcm(AP,BP'): =: lcm(o(X),o(Y)).$






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            No. For instance if $G=mathbb Z_2oplusmathbb Z_2$, then $|G|=4$, but the least common multiple of the order of it's group elements is $2$.






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              2 Answers
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              2 Answers
              2






              active

              oldest

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              active

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              active

              oldest

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              0














              No, for a finite abelian group the $rm lcm$ of the orders is the expt (not order) of the group, namely



              $begin{eqnarray}rm{bf Proposition}quad maxord(G) !&,=,&rm expt(G) text{ for a finite abelian group} G, i.e.\ \
              rm max { ord(g) : : g in G} !&=&rm min { n>0 : : g^n = 1 forall g in G}end{eqnarray}$



              Proof $ $ By the lemma below, $rm: S, =, { ord(g) : :g in G }$ is a finite
              set of naturals closed under$rm lcm$.



              Hence every $rm s in S:$ is a
              divisor of the max elt $rm: m $ [else $rm: lcm(s,m) > m,$],$ $ so $rm m = expt(G)$.



              Lemma $ $ A finite abelian group $rm:G:$ has an lcm-closed order set, i.e. with $rm: o(X) = $ order of $rm: X$



              $$rm X,Y in G Rightarrow exists Z in G: o(Z) = lcm(o(X),o(Y))$$



              Proof $ $ By induction on $rm: o(X), o(Y). $ If it's $:1:$ then trivially $rm:Z = 1$. $ $ Otherwise



              write $rm o(X) =: AP,: o(Y) = BP', P'|,P = p^m > 1, $ prime $rm: p:$ coprime to $rm: A,B.$



              Then $rm: o(X^P) = A, o(Y^{P'}) = B. $ By induction there's a $rm: Z:$ with $rm : o(Z) = lcm(A,B)$



              so $rm o(X^A: Z): =: P lcm(A,B): =: lcm(AP,BP'): =: lcm(o(X),o(Y)).$






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                No, for a finite abelian group the $rm lcm$ of the orders is the expt (not order) of the group, namely



                $begin{eqnarray}rm{bf Proposition}quad maxord(G) !&,=,&rm expt(G) text{ for a finite abelian group} G, i.e.\ \
                rm max { ord(g) : : g in G} !&=&rm min { n>0 : : g^n = 1 forall g in G}end{eqnarray}$



                Proof $ $ By the lemma below, $rm: S, =, { ord(g) : :g in G }$ is a finite
                set of naturals closed under$rm lcm$.



                Hence every $rm s in S:$ is a
                divisor of the max elt $rm: m $ [else $rm: lcm(s,m) > m,$],$ $ so $rm m = expt(G)$.



                Lemma $ $ A finite abelian group $rm:G:$ has an lcm-closed order set, i.e. with $rm: o(X) = $ order of $rm: X$



                $$rm X,Y in G Rightarrow exists Z in G: o(Z) = lcm(o(X),o(Y))$$



                Proof $ $ By induction on $rm: o(X), o(Y). $ If it's $:1:$ then trivially $rm:Z = 1$. $ $ Otherwise



                write $rm o(X) =: AP,: o(Y) = BP', P'|,P = p^m > 1, $ prime $rm: p:$ coprime to $rm: A,B.$



                Then $rm: o(X^P) = A, o(Y^{P'}) = B. $ By induction there's a $rm: Z:$ with $rm : o(Z) = lcm(A,B)$



                so $rm o(X^A: Z): =: P lcm(A,B): =: lcm(AP,BP'): =: lcm(o(X),o(Y)).$






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                  No, for a finite abelian group the $rm lcm$ of the orders is the expt (not order) of the group, namely



                  $begin{eqnarray}rm{bf Proposition}quad maxord(G) !&,=,&rm expt(G) text{ for a finite abelian group} G, i.e.\ \
                  rm max { ord(g) : : g in G} !&=&rm min { n>0 : : g^n = 1 forall g in G}end{eqnarray}$



                  Proof $ $ By the lemma below, $rm: S, =, { ord(g) : :g in G }$ is a finite
                  set of naturals closed under$rm lcm$.



                  Hence every $rm s in S:$ is a
                  divisor of the max elt $rm: m $ [else $rm: lcm(s,m) > m,$],$ $ so $rm m = expt(G)$.



                  Lemma $ $ A finite abelian group $rm:G:$ has an lcm-closed order set, i.e. with $rm: o(X) = $ order of $rm: X$



                  $$rm X,Y in G Rightarrow exists Z in G: o(Z) = lcm(o(X),o(Y))$$



                  Proof $ $ By induction on $rm: o(X), o(Y). $ If it's $:1:$ then trivially $rm:Z = 1$. $ $ Otherwise



                  write $rm o(X) =: AP,: o(Y) = BP', P'|,P = p^m > 1, $ prime $rm: p:$ coprime to $rm: A,B.$



                  Then $rm: o(X^P) = A, o(Y^{P'}) = B. $ By induction there's a $rm: Z:$ with $rm : o(Z) = lcm(A,B)$



                  so $rm o(X^A: Z): =: P lcm(A,B): =: lcm(AP,BP'): =: lcm(o(X),o(Y)).$






                  share|cite|improve this answer












                  No, for a finite abelian group the $rm lcm$ of the orders is the expt (not order) of the group, namely



                  $begin{eqnarray}rm{bf Proposition}quad maxord(G) !&,=,&rm expt(G) text{ for a finite abelian group} G, i.e.\ \
                  rm max { ord(g) : : g in G} !&=&rm min { n>0 : : g^n = 1 forall g in G}end{eqnarray}$



                  Proof $ $ By the lemma below, $rm: S, =, { ord(g) : :g in G }$ is a finite
                  set of naturals closed under$rm lcm$.



                  Hence every $rm s in S:$ is a
                  divisor of the max elt $rm: m $ [else $rm: lcm(s,m) > m,$],$ $ so $rm m = expt(G)$.



                  Lemma $ $ A finite abelian group $rm:G:$ has an lcm-closed order set, i.e. with $rm: o(X) = $ order of $rm: X$



                  $$rm X,Y in G Rightarrow exists Z in G: o(Z) = lcm(o(X),o(Y))$$



                  Proof $ $ By induction on $rm: o(X), o(Y). $ If it's $:1:$ then trivially $rm:Z = 1$. $ $ Otherwise



                  write $rm o(X) =: AP,: o(Y) = BP', P'|,P = p^m > 1, $ prime $rm: p:$ coprime to $rm: A,B.$



                  Then $rm: o(X^P) = A, o(Y^{P'}) = B. $ By induction there's a $rm: Z:$ with $rm : o(Z) = lcm(A,B)$



                  so $rm o(X^A: Z): =: P lcm(A,B): =: lcm(AP,BP'): =: lcm(o(X),o(Y)).$







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                  answered Nov 26 at 3:39









                  Bill Dubuque

                  208k29190626




                  208k29190626























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                      No. For instance if $G=mathbb Z_2oplusmathbb Z_2$, then $|G|=4$, but the least common multiple of the order of it's group elements is $2$.






                      share|cite|improve this answer


























                        1














                        No. For instance if $G=mathbb Z_2oplusmathbb Z_2$, then $|G|=4$, but the least common multiple of the order of it's group elements is $2$.






                        share|cite|improve this answer
























                          1












                          1








                          1






                          No. For instance if $G=mathbb Z_2oplusmathbb Z_2$, then $|G|=4$, but the least common multiple of the order of it's group elements is $2$.






                          share|cite|improve this answer












                          No. For instance if $G=mathbb Z_2oplusmathbb Z_2$, then $|G|=4$, but the least common multiple of the order of it's group elements is $2$.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Nov 26 at 3:27









                          Aweygan

                          13.4k21441




                          13.4k21441















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