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YCor
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Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$. Let $V$ be a nontrivial irreducible $\mathbb{F}_p[G]$ module. I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$-part of $|G|$). By checking the Atlas of Finite GroupGroups, $|V|$ sometimes lesser than $|G|_p$ and sometimes larger than $|G|_p$. For example, if $|G|_p=p$ and $V$ is any nontrivial irreducible $\mathbb{F}_p[G]$-module, then $|V|>|G|_p$, if $G=SL_4(2)$$G=\mathrm{SL}_4(2)$ and $V$ is a nontrivial irreducible $\mathbb{F}_2[G]$-module of dimension 4, then $|V|<|G|_2$.

Are there some inequalities to show the relation of $|V|$ and $|G|_p$?
For instance, could $|G|_p$ be bounded by a function of $|V|$? when $|V|\mid |G|$$|V|$ divides $|G|$?

Any explanation, references suggestion and examples are appreciated.

Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$. Let $V$ be a nontrivial irreducible $\mathbb{F}_p[G]$ module. I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$-part of $|G|$). By checking the Atlas of Finite Group, $|V|$ sometimes lesser than $|G|_p$ and sometimes larger than $|G|_p$. For example, if $|G|_p=p$ and $V$ is any nontrivial irreducible $\mathbb{F}_p[G]$-module, then $|V|>|G|_p$, if $G=SL_4(2)$ and $V$ is a nontrivial irreducible $\mathbb{F}_2[G]$-module of dimension 4, then $|V|<|G|_2$.

Are there some inequalities to show the relation of $|V|$ and $|G|_p$?
For instance, could $|G|_p$ be bounded by a function of $|V|$? when $|V|\mid |G|$?

Any explanation, references suggestion and examples are appreciated.

Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$. Let $V$ be a nontrivial irreducible $\mathbb{F}_p[G]$ module. I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$-part of $|G|$). By checking the Atlas of Finite Groups, $|V|$ sometimes lesser than $|G|_p$ and sometimes larger than $|G|_p$. For example, if $|G|_p=p$ and $V$ is any nontrivial irreducible $\mathbb{F}_p[G]$-module, then $|V|>|G|_p$, if $G=\mathrm{SL}_4(2)$ and $V$ is a nontrivial irreducible $\mathbb{F}_2[G]$-module of dimension 4, then $|V|<|G|_2$.

Are there some inequalities to show the relation of $|V|$ and $|G|_p$?
For instance, could $|G|_p$ be bounded by a function of $|V|$? when $|V|$ divides $|G|$?

Any explanation, references suggestion and examples are appreciated.

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user44312
user44312
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Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$. Let $V$ be ana nontrivial irreducible $\mathbb{F}_p[G]$ module. I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$-part of $|G|$). By checking the Atlas of Finite Group, $|V|$ sometimes lesser than $|G|_p$ and sometimes larger than $|G|_p$. For example, if $|G|_p=p$ and $V$ is anany nontrivial irreducible $\mathbb{F}_p[G]$-module, then $|V|>|G|_p$;, if $G=PSL_4(2)$$G=SL_4(2)$ and $V$ is ana nontrivial irreducible $\mathbb{F}_2[G]$-module of dimension 4, then $|V|<|G|_2$.

Are there some inequalities to show the relation of $|V|$ and $|G|_p$?
For instance, could $|G|_p$ be bounded by a function of $|V|$? when $|V|\mid |G|$?

Any explanation, references suggestion and examples are appreciated.

Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$. Let $V$ be an irreducible $\mathbb{F}_p[G]$ module. I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$-part of $|G|$). By checking the Atlas of Finite Group, $|V|$ sometimes lesser than $|G|_p$ and sometimes larger than $|G|_p$. For example, if $|G|_p=p$ and $V$ is an irreducible $\mathbb{F}_p[G]$-module, then $|V|>|G|_p$; if $G=PSL_4(2)$ and $V$ is an irreducible $\mathbb{F}_2[G]$-module of dimension 4, then $|V|<|G|_2$.

Are there some inequalities to show the relation of $|V|$ and $|G|_p$?
For instance, could $|G|_p$ be bounded by a function of $|V|$?

Any explanation, references suggestion and examples are appreciated.

Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$. Let $V$ be a nontrivial irreducible $\mathbb{F}_p[G]$ module. I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$-part of $|G|$). By checking the Atlas of Finite Group, $|V|$ sometimes lesser than $|G|_p$ and sometimes larger than $|G|_p$. For example, if $|G|_p=p$ and $V$ is any nontrivial irreducible $\mathbb{F}_p[G]$-module, then $|V|>|G|_p$, if $G=SL_4(2)$ and $V$ is a nontrivial irreducible $\mathbb{F}_2[G]$-module of dimension 4, then $|V|<|G|_2$.

Are there some inequalities to show the relation of $|V|$ and $|G|_p$?
For instance, could $|G|_p$ be bounded by a function of $|V|$? when $|V|\mid |G|$?

Any explanation, references suggestion and examples are appreciated.

Source Link
user44312
user44312
  • 613
  • 3
  • 7

Irreducible module of finite simple groups

Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$. Let $V$ be an irreducible $\mathbb{F}_p[G]$ module. I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$-part of $|G|$). By checking the Atlas of Finite Group, $|V|$ sometimes lesser than $|G|_p$ and sometimes larger than $|G|_p$. For example, if $|G|_p=p$ and $V$ is an irreducible $\mathbb{F}_p[G]$-module, then $|V|>|G|_p$; if $G=PSL_4(2)$ and $V$ is an irreducible $\mathbb{F}_2[G]$-module of dimension 4, then $|V|<|G|_2$.

Are there some inequalities to show the relation of $|V|$ and $|G|_p$?
For instance, could $|G|_p$ be bounded by a function of $|V|$?

Any explanation, references suggestion and examples are appreciated.