Representation of a finite group and its Sylow $p$-subgroup
0
$begingroup$
Let $G$ be a finite group with order $|G|=p^n cdot m$ for some positive integers $n,m$ and $H$ be a Sylow $p$-subgroup of $G$. What relations can we say about the representations of $G$ and $H$?
finite-groups representation-theory sylow-theory
edited Apr 12 '16 at 10:59
Andreas Caranti
56.9k34397
asked Apr 12 '16 at 6:44
NinjaNinja
1,115720
$endgroup$
|
show 1 more comment
0
$begingroup$
Let $G$ be a finite group with order $|G|=p^n cdot m$ for some positive integers $n,m$ and $H$ be a Sylow $p$-subgroup of $G$. What relations can we say about the representations of $G$ and $H$?
finite-groups representation-theory sylow-theory
edited Apr 12 '16 at 10:59
Andreas Caranti
56.9k34397
asked Apr 12 '16 at 6:44
NinjaNinja
1,115720
$endgroup$
1
$begingroup$
I don't understand the question. What does "are they equivalent representations" mean?
$endgroup$
– Qiaochu Yuan
Apr 12 '16 at 6:45
$begingroup$
"equivalent representations" was just an example. By equivalent representations, I mean the following: If there are representations $rho_1:G rightarrow GL(n,mathbb{F})$ and $rho_2:G rightarrow GL(n,mathbb{F})$ we say that they are equivalent if there exists an element $T in GL(n,mathbb{F})$ such that $rho_{1}(g)=T.rho_{2}(g)T^{-1}$
$endgroup$
– Ninja
Apr 12 '16 at 6:53
$begingroup$
Oh, it seems like my example is wrong since equivalent representations are the representations of the same group. I'm editing the question.
$endgroup$
– Ninja
Apr 12 '16 at 6:58
$begingroup$
Your edited question still does not make sense. What does conjugation do to a map? And how could this make it a map with a different domain?
$endgroup$
– Tobias Kildetoft
Apr 12 '16 at 7:07
$begingroup$
Okay, I totally deleted my example, I will give a clear example if we can't get any answers. My bad.
$endgroup$
– Ninja
Apr 12 '16 at 7:11
|
show 1 more comment
0
0
0
$begingroup$
Let $G$ be a finite group with order $|G|=p^n cdot m$ for some positive integers $n,m$ and $H$ be a Sylow $p$-subgroup of $G$. What relations can we say about the representations of $G$ and $H$?
finite-groups representation-theory sylow-theory
edited Apr 12 '16 at 10:59
Andreas Caranti
56.9k34397
asked Apr 12 '16 at 6:44
NinjaNinja
1,115720
$endgroup$
Let $G$ be a finite group with order $|G|=p^n cdot m$ for some positive integers $n,m$ and $H$ be a Sylow $p$-subgroup of $G$. What relations can we say about the representations of $G$ and $H$?
finite-groups representation-theory sylow-theory
finite-groups representation-theory sylow-theory
edited Apr 12 '16 at 10:59
Andreas Caranti
56.9k34397
asked Apr 12 '16 at 6:44
NinjaNinja
1,115720
edited Apr 12 '16 at 10:59
Andreas Caranti
56.9k34397
asked Apr 12 '16 at 6:44
NinjaNinja
1,115720
edited Apr 12 '16 at 10:59
Andreas Caranti
56.9k34397
edited Apr 12 '16 at 10:59
Andreas Caranti
56.9k34397
edited Apr 12 '16 at 10:59
Andreas Caranti
56.9k34397
56.9k34397
asked Apr 12 '16 at 6:44
NinjaNinja
1,115720
asked Apr 12 '16 at 6:44
NinjaNinja
1,115720
asked Apr 12 '16 at 6:44
NinjaNinja
1,115720
1,115720
1
$begingroup$
I don't understand the question. What does "are they equivalent representations" mean?
$endgroup$
– Qiaochu Yuan
Apr 12 '16 at 6:45
$begingroup$
"equivalent representations" was just an example. By equivalent representations, I mean the following: If there are representations $rho_1:G rightarrow GL(n,mathbb{F})$ and $rho_2:G rightarrow GL(n,mathbb{F})$ we say that they are equivalent if there exists an element $T in GL(n,mathbb{F})$ such that $rho_{1}(g)=T.rho_{2}(g)T^{-1}$
$endgroup$
– Ninja
Apr 12 '16 at 6:53
$begingroup$
Oh, it seems like my example is wrong since equivalent representations are the representations of the same group. I'm editing the question.
$endgroup$
– Ninja
Apr 12 '16 at 6:58
$begingroup$
Your edited question still does not make sense. What does conjugation do to a map? And how could this make it a map with a different domain?
$endgroup$
– Tobias Kildetoft
Apr 12 '16 at 7:07
$begingroup$
Okay, I totally deleted my example, I will give a clear example if we can't get any answers. My bad.
$endgroup$
– Ninja
Apr 12 '16 at 7:11
|
show 1 more comment
1
$begingroup$
I don't understand the question. What does "are they equivalent representations" mean?
$endgroup$
– Qiaochu Yuan
Apr 12 '16 at 6:45
$begingroup$
"equivalent representations" was just an example. By equivalent representations, I mean the following: If there are representations $rho_1:G rightarrow GL(n,mathbb{F})$ and $rho_2:G rightarrow GL(n,mathbb{F})$ we say that they are equivalent if there exists an element $T in GL(n,mathbb{F})$ such that $rho_{1}(g)=T.rho_{2}(g)T^{-1}$
$endgroup$
– Ninja
Apr 12 '16 at 6:53
$begingroup$
Oh, it seems like my example is wrong since equivalent representations are the representations of the same group. I'm editing the question.
$endgroup$
– Ninja
Apr 12 '16 at 6:58
$begingroup$
Your edited question still does not make sense. What does conjugation do to a map? And how could this make it a map with a different domain?
$endgroup$
– Tobias Kildetoft
Apr 12 '16 at 7:07
$begingroup$
Okay, I totally deleted my example, I will give a clear example if we can't get any answers. My bad.
$endgroup$
– Ninja
Apr 12 '16 at 7:11
1
1
$begingroup$
I don't understand the question. What does "are they equivalent representations" mean?
$endgroup$
– Qiaochu Yuan
Apr 12 '16 at 6:45
$begingroup$
I don't understand the question. What does "are they equivalent representations" mean?
$endgroup$
– Qiaochu Yuan
Apr 12 '16 at 6:45
$begingroup$
"equivalent representations" was just an example. By equivalent representations, I mean the following: If there are representations $rho_1:G rightarrow GL(n,mathbb{F})$ and $rho_2:G rightarrow GL(n,mathbb{F})$ we say that they are equivalent if there exists an element $T in GL(n,mathbb{F})$ such that $rho_{1}(g)=T.rho_{2}(g)T^{-1}$
$endgroup$
– Ninja
Apr 12 '16 at 6:53
$begingroup$
"equivalent representations" was just an example. By equivalent representations, I mean the following: If there are representations $rho_1:G rightarrow GL(n,mathbb{F})$ and $rho_2:G rightarrow GL(n,mathbb{F})$ we say that they are equivalent if there exists an element $T in GL(n,mathbb{F})$ such that $rho_{1}(g)=T.rho_{2}(g)T^{-1}$
$endgroup$
– Ninja
Apr 12 '16 at 6:53
$begingroup$
Oh, it seems like my example is wrong since equivalent representations are the representations of the same group. I'm editing the question.
$endgroup$
– Ninja
Apr 12 '16 at 6:58
$begingroup$
Oh, it seems like my example is wrong since equivalent representations are the representations of the same group. I'm editing the question.
$endgroup$
– Ninja
Apr 12 '16 at 6:58
$begingroup$
Your edited question still does not make sense. What does conjugation do to a map? And how could this make it a map with a different domain?
$endgroup$
– Tobias Kildetoft
Apr 12 '16 at 7:07
$begingroup$
Your edited question still does not make sense. What does conjugation do to a map? And how could this make it a map with a different domain?
$endgroup$
– Tobias Kildetoft
Apr 12 '16 at 7:07
$begingroup$
Okay, I totally deleted my example, I will give a clear example if we can't get any answers. My bad.
$endgroup$
– Ninja
Apr 12 '16 at 7:11
$begingroup$
Okay, I totally deleted my example, I will give a clear example if we can't get any answers. My bad.
$endgroup$
– Ninja
Apr 12 '16 at 7:11
|
show 1 more comment
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1
$begingroup$
I don't understand the question. What does "are they equivalent representations" mean?
$endgroup$
– Qiaochu Yuan
Apr 12 '16 at 6:45
$begingroup$
"equivalent representations" was just an example. By equivalent representations, I mean the following: If there are representations $rho_1:G rightarrow GL(n,mathbb{F})$ and $rho_2:G rightarrow GL(n,mathbb{F})$ we say that they are equivalent if there exists an element $T in GL(n,mathbb{F})$ such that $rho_{1}(g)=T.rho_{2}(g)T^{-1}$
$endgroup$
– Ninja
Apr 12 '16 at 6:53
$begingroup$
Oh, it seems like my example is wrong since equivalent representations are the representations of the same group. I'm editing the question.
$endgroup$
– Ninja
Apr 12 '16 at 6:58
$begingroup$
Your edited question still does not make sense. What does conjugation do to a map? And how could this make it a map with a different domain?
$endgroup$
– Tobias Kildetoft
Apr 12 '16 at 7:07
$begingroup$
Okay, I totally deleted my example, I will give a clear example if we can't get any answers. My bad.
$endgroup$
– Ninja
Apr 12 '16 at 7:11