What does $O_p(G)$ mean in the context of a Fitting subgroup of $G$?












1












$begingroup$


I asked this question as a comment on this old, probably abandoned question about the Fitting subgroup $F(G)$ of a group $G$.



It was stated in the question that




$F(G)$ is the product of all $O_p(G)$ for all prime $p$.




(Edit: Derek Holt pointed out below that $G$ must be finite here.)



I'm just curious. I couldn't find the notation in any of the obvious places, like in the results of a simple Google search. I'm interested in the theorem quoted above.



The closest thing I've seen before is the notation for the ring of integers in the context of algebraic number theory, which is way off.






group-theory notation direct-product







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    en.wikipedia.org/wiki/Core_(group_theory)#The_p-core
    $endgroup$
    – Lord Shark the Unknown
    Dec 22 '18 at 6:23










  • $begingroup$
    Thank you, @LordSharktheUnknown; please would you elaborate on that to make it an answer so that I may close this question?
    $endgroup$
    – Shaun
    Dec 22 '18 at 6:25






  • 1




    $begingroup$
    By the way, the result in question applies only to finite groups. For general groups $F(G)$ is not always defined.
    $endgroup$
    – Derek Holt
    Dec 22 '18 at 9:22
















1












$begingroup$


I asked this question as a comment on this old, probably abandoned question about the Fitting subgroup $F(G)$ of a group $G$.



It was stated in the question that




$F(G)$ is the product of all $O_p(G)$ for all prime $p$.




(Edit: Derek Holt pointed out below that $G$ must be finite here.)



I'm just curious. I couldn't find the notation in any of the obvious places, like in the results of a simple Google search. I'm interested in the theorem quoted above.



The closest thing I've seen before is the notation for the ring of integers in the context of algebraic number theory, which is way off.






group-theory notation direct-product







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    en.wikipedia.org/wiki/Core_(group_theory)#The_p-core
    $endgroup$
    – Lord Shark the Unknown
    Dec 22 '18 at 6:23










  • $begingroup$
    Thank you, @LordSharktheUnknown; please would you elaborate on that to make it an answer so that I may close this question?
    $endgroup$
    – Shaun
    Dec 22 '18 at 6:25






  • 1




    $begingroup$
    By the way, the result in question applies only to finite groups. For general groups $F(G)$ is not always defined.
    $endgroup$
    – Derek Holt
    Dec 22 '18 at 9:22














1












1








1


1



$begingroup$


I asked this question as a comment on this old, probably abandoned question about the Fitting subgroup $F(G)$ of a group $G$.



It was stated in the question that




$F(G)$ is the product of all $O_p(G)$ for all prime $p$.




(Edit: Derek Holt pointed out below that $G$ must be finite here.)



I'm just curious. I couldn't find the notation in any of the obvious places, like in the results of a simple Google search. I'm interested in the theorem quoted above.



The closest thing I've seen before is the notation for the ring of integers in the context of algebraic number theory, which is way off.






group-theory notation direct-product







share|cite|improve this question











$endgroup$




I asked this question as a comment on this old, probably abandoned question about the Fitting subgroup $F(G)$ of a group $G$.



It was stated in the question that




$F(G)$ is the product of all $O_p(G)$ for all prime $p$.




(Edit: Derek Holt pointed out below that $G$ must be finite here.)



I'm just curious. I couldn't find the notation in any of the obvious places, like in the results of a simple Google search. I'm interested in the theorem quoted above.



The closest thing I've seen before is the notation for the ring of integers in the context of algebraic number theory, which is way off.






group-theory notation direct-product




group-theory notation direct-product






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 22 '18 at 9:24







Shaun

















asked Dec 22 '18 at 6:18









ShaunShaun

9,380113684




9,380113684








  • 1




    $begingroup$
    en.wikipedia.org/wiki/Core_(group_theory)#The_p-core
    $endgroup$
    – Lord Shark the Unknown
    Dec 22 '18 at 6:23










  • $begingroup$
    Thank you, @LordSharktheUnknown; please would you elaborate on that to make it an answer so that I may close this question?
    $endgroup$
    – Shaun
    Dec 22 '18 at 6:25






  • 1




    $begingroup$
    By the way, the result in question applies only to finite groups. For general groups $F(G)$ is not always defined.
    $endgroup$
    – Derek Holt
    Dec 22 '18 at 9:22














  • 1




    $begingroup$
    en.wikipedia.org/wiki/Core_(group_theory)#The_p-core
    $endgroup$
    – Lord Shark the Unknown
    Dec 22 '18 at 6:23










  • $begingroup$
    Thank you, @LordSharktheUnknown; please would you elaborate on that to make it an answer so that I may close this question?
    $endgroup$
    – Shaun
    Dec 22 '18 at 6:25






  • 1




    $begingroup$
    By the way, the result in question applies only to finite groups. For general groups $F(G)$ is not always defined.
    $endgroup$
    – Derek Holt
    Dec 22 '18 at 9:22








1




1




$begingroup$
en.wikipedia.org/wiki/Core_(group_theory)#The_p-core
$endgroup$
– Lord Shark the Unknown
Dec 22 '18 at 6:23




$begingroup$
en.wikipedia.org/wiki/Core_(group_theory)#The_p-core
$endgroup$
– Lord Shark the Unknown
Dec 22 '18 at 6:23












$begingroup$
Thank you, @LordSharktheUnknown; please would you elaborate on that to make it an answer so that I may close this question?
$endgroup$
– Shaun
Dec 22 '18 at 6:25




$begingroup$
Thank you, @LordSharktheUnknown; please would you elaborate on that to make it an answer so that I may close this question?
$endgroup$
– Shaun
Dec 22 '18 at 6:25




1




1




$begingroup$
By the way, the result in question applies only to finite groups. For general groups $F(G)$ is not always defined.
$endgroup$
– Derek Holt
Dec 22 '18 at 9:22




$begingroup$
By the way, the result in question applies only to finite groups. For general groups $F(G)$ is not always defined.
$endgroup$
– Derek Holt
Dec 22 '18 at 9:22










1 Answer
1






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oldest

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2












$begingroup$

$O_p(G)$ is the $p$-core of the finite group $G$. This is the intersection of all its Sylow $p$-subgroups, and is the largest normal $p$-subgroup of $G$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    This makes me wish of a group $C$ whose name sounds like "choo", just so we could have $O_p(C)$.
    $endgroup$
    – Shaun
    Dec 22 '18 at 6:44













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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

$O_p(G)$ is the $p$-core of the finite group $G$. This is the intersection of all its Sylow $p$-subgroups, and is the largest normal $p$-subgroup of $G$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    This makes me wish of a group $C$ whose name sounds like "choo", just so we could have $O_p(C)$.
    $endgroup$
    – Shaun
    Dec 22 '18 at 6:44


















2












$begingroup$

$O_p(G)$ is the $p$-core of the finite group $G$. This is the intersection of all its Sylow $p$-subgroups, and is the largest normal $p$-subgroup of $G$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    This makes me wish of a group $C$ whose name sounds like "choo", just so we could have $O_p(C)$.
    $endgroup$
    – Shaun
    Dec 22 '18 at 6:44
















2












2








2





$begingroup$

$O_p(G)$ is the $p$-core of the finite group $G$. This is the intersection of all its Sylow $p$-subgroups, and is the largest normal $p$-subgroup of $G$.






share|cite|improve this answer











$endgroup$



$O_p(G)$ is the $p$-core of the finite group $G$. This is the intersection of all its Sylow $p$-subgroups, and is the largest normal $p$-subgroup of $G$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 22 '18 at 6:40









Shaun

9,380113684




9,380113684










answered Dec 22 '18 at 6:28









Lord Shark the UnknownLord Shark the Unknown

105k1160133




105k1160133












  • $begingroup$
    This makes me wish of a group $C$ whose name sounds like "choo", just so we could have $O_p(C)$.
    $endgroup$
    – Shaun
    Dec 22 '18 at 6:44




















  • $begingroup$
    This makes me wish of a group $C$ whose name sounds like "choo", just so we could have $O_p(C)$.
    $endgroup$
    – Shaun
    Dec 22 '18 at 6:44


















$begingroup$
This makes me wish of a group $C$ whose name sounds like "choo", just so we could have $O_p(C)$.
$endgroup$
– Shaun
Dec 22 '18 at 6:44






$begingroup$
This makes me wish of a group $C$ whose name sounds like "choo", just so we could have $O_p(C)$.
$endgroup$
– Shaun
Dec 22 '18 at 6:44




















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