Let $G$ be a non-abelian finite $p$-group, do we have $Z(G) leq Phi(G)$ in general?
Let $G$ be a non-abelian finite $p$-group. I wonder if $Z(G) leq Phi(G)$, where $Phi(G)$ is the Frattini subgroup of $G$? I'd known that it is true for non-abelian groups of order $p^3$ but I doubt it is true in general or not.
I understand that it isn't true in general. As showed below it isn't true for some group of order $p^4$.
group-theory finite-groups normal-subgroups p-groups maximal-subgroup
asked 2 days ago
Yasmin
13511
|
show 2 more comments
Let $G$ be a non-abelian finite $p$-group. I wonder if $Z(G) leq Phi(G)$, where $Phi(G)$ is the Frattini subgroup of $G$? I'd known that it is true for non-abelian groups of order $p^3$ but I doubt it is true in general or not.
I understand that it isn't true in general. As showed below it isn't true for some group of order $p^4$.
group-theory finite-groups normal-subgroups p-groups maximal-subgroup
asked 2 days ago
Yasmin
13511
@Yasmin: Your edit does not address the original problem: why did you think it was true, or why did you want to prove it? As to what you write now, you “ask” whether it is true, and then you say you “understand” it isn’t. That makes the question self-contradictory.
– Arturo Magidin
2 days ago
@ArturoMagidin Would you please suggest how to edit it in a right way?
– Yasmin
2 days ago
@Yasmin: Add context. Add why you were interested in this; why you thought it might be true. Do not falsify the record by writing nonsense after the fact when it turned out that what you wanted to prove is actually false. That would be a good place to start, instead of asking people to do your work for you, again.
– Arturo Magidin
2 days ago
2
The general point is that if you do not know whether something is true or not, then you should not say "help me prove this", you should say "help me decide whether this is true". Otherwise people might waste time trying to prove something that isn't true. You should also say where the problem comes from: is it a homework exercise, is it a problem from a book, is it your own problem? (Occasionally problems from books are wrong, and of course in that case everybody would be confused!)
– Derek Holt
2 days ago
1
I have voted to reopen, because the question itself has some interest and it has been correctly answered.
– Derek Holt
2 days ago
|
show 2 more comments
Let $G$ be a non-abelian finite $p$-group. I wonder if $Z(G) leq Phi(G)$, where $Phi(G)$ is the Frattini subgroup of $G$? I'd known that it is true for non-abelian groups of order $p^3$ but I doubt it is true in general or not.
I understand that it isn't true in general. As showed below it isn't true for some group of order $p^4$.
group-theory finite-groups normal-subgroups p-groups maximal-subgroup
asked 2 days ago
Yasmin
13511
Let $G$ be a non-abelian finite $p$-group. I wonder if $Z(G) leq Phi(G)$, where $Phi(G)$ is the Frattini subgroup of $G$? I'd known that it is true for non-abelian groups of order $p^3$ but I doubt it is true in general or not.
I understand that it isn't true in general. As showed below it isn't true for some group of order $p^4$.
group-theory finite-groups normal-subgroups p-groups maximal-subgroup
group-theory finite-groups normal-subgroups p-groups maximal-subgroup
asked 2 days ago
Yasmin
13511
asked 2 days ago
Yasmin
13511
edited 2 days ago
asked 2 days ago
Yasmin
13511
asked 2 days ago
Yasmin
13511
asked 2 days ago
Yasmin
13511
13511
@Yasmin: Your edit does not address the original problem: why did you think it was true, or why did you want to prove it? As to what you write now, you “ask” whether it is true, and then you say you “understand” it isn’t. That makes the question self-contradictory.
– Arturo Magidin
2 days ago
@ArturoMagidin Would you please suggest how to edit it in a right way?
– Yasmin
2 days ago
@Yasmin: Add context. Add why you were interested in this; why you thought it might be true. Do not falsify the record by writing nonsense after the fact when it turned out that what you wanted to prove is actually false. That would be a good place to start, instead of asking people to do your work for you, again.
– Arturo Magidin
2 days ago
2
The general point is that if you do not know whether something is true or not, then you should not say "help me prove this", you should say "help me decide whether this is true". Otherwise people might waste time trying to prove something that isn't true. You should also say where the problem comes from: is it a homework exercise, is it a problem from a book, is it your own problem? (Occasionally problems from books are wrong, and of course in that case everybody would be confused!)
– Derek Holt
2 days ago
1
I have voted to reopen, because the question itself has some interest and it has been correctly answered.
– Derek Holt
2 days ago
|
show 2 more comments
@Yasmin: Your edit does not address the original problem: why did you think it was true, or why did you want to prove it? As to what you write now, you “ask” whether it is true, and then you say you “understand” it isn’t. That makes the question self-contradictory.
– Arturo Magidin
2 days ago
@ArturoMagidin Would you please suggest how to edit it in a right way?
– Yasmin
2 days ago
@Yasmin: Add context. Add why you were interested in this; why you thought it might be true. Do not falsify the record by writing nonsense after the fact when it turned out that what you wanted to prove is actually false. That would be a good place to start, instead of asking people to do your work for you, again.
– Arturo Magidin
2 days ago
2
The general point is that if you do not know whether something is true or not, then you should not say "help me prove this", you should say "help me decide whether this is true". Otherwise people might waste time trying to prove something that isn't true. You should also say where the problem comes from: is it a homework exercise, is it a problem from a book, is it your own problem? (Occasionally problems from books are wrong, and of course in that case everybody would be confused!)
– Derek Holt
2 days ago
1
I have voted to reopen, because the question itself has some interest and it has been correctly answered.
– Derek Holt
2 days ago
@Yasmin: Your edit does not address the original problem: why did you think it was true, or why did you want to prove it? As to what you write now, you “ask” whether it is true, and then you say you “understand” it isn’t. That makes the question self-contradictory.
– Arturo Magidin
2 days ago
@Yasmin: Your edit does not address the original problem: why did you think it was true, or why did you want to prove it? As to what you write now, you “ask” whether it is true, and then you say you “understand” it isn’t. That makes the question self-contradictory.
– Arturo Magidin
2 days ago
@ArturoMagidin Would you please suggest how to edit it in a right way?
– Yasmin
2 days ago
@ArturoMagidin Would you please suggest how to edit it in a right way?
– Yasmin
2 days ago
@Yasmin: Add context. Add why you were interested in this; why you thought it might be true. Do not falsify the record by writing nonsense after the fact when it turned out that what you wanted to prove is actually false. That would be a good place to start, instead of asking people to do your work for you, again.
– Arturo Magidin
2 days ago
@Yasmin: Add context. Add why you were interested in this; why you thought it might be true. Do not falsify the record by writing nonsense after the fact when it turned out that what you wanted to prove is actually false. That would be a good place to start, instead of asking people to do your work for you, again.
– Arturo Magidin
2 days ago
2
2
The general point is that if you do not know whether something is true or not, then you should not say "help me prove this", you should say "help me decide whether this is true". Otherwise people might waste time trying to prove something that isn't true. You should also say where the problem comes from: is it a homework exercise, is it a problem from a book, is it your own problem? (Occasionally problems from books are wrong, and of course in that case everybody would be confused!)
– Derek Holt
2 days ago
The general point is that if you do not know whether something is true or not, then you should not say "help me prove this", you should say "help me decide whether this is true". Otherwise people might waste time trying to prove something that isn't true. You should also say where the problem comes from: is it a homework exercise, is it a problem from a book, is it your own problem? (Occasionally problems from books are wrong, and of course in that case everybody would be confused!)
– Derek Holt
2 days ago
1
1
I have voted to reopen, because the question itself has some interest and it has been correctly answered.
– Derek Holt
2 days ago
I have voted to reopen, because the question itself has some interest and it has been correctly answered.
– Derek Holt
2 days ago
|
show 2 more comments
1 Answer
1
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Let $H$ be the nonabelian group of order $p^3$ and exponent $p$. Let $C_p$ be a cyclic group of order $p$. Let $G=Htimes C_p$. Then $Z(G)$ is a direct product of two copies of $C_p$, one contained in $H$, the other the second direct factor of $G$.
However, $H$ is maximal in $G$, and so $Phi(G)subseteq H$. Thus, $Z(G)$ is not contained in $Phi(G)$.
More generally, given any nonabelian $p$-group $K$, letting $G=Ktimes C_p$ gives you a group with center $Z(G) = Z(K)times C_p$, and with $K$ maximal and hence $Phi(G)subseteq K$, hence a counterexample to the assertion to $Z(G)subseteq Phi(G)$.
answered 2 days ago
Arturo Magidin
261k32585906
1
Thank you so much.
– Yasmin
2 days ago
add a comment |
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Let $H$ be the nonabelian group of order $p^3$ and exponent $p$. Let $C_p$ be a cyclic group of order $p$. Let $G=Htimes C_p$. Then $Z(G)$ is a direct product of two copies of $C_p$, one contained in $H$, the other the second direct factor of $G$.
However, $H$ is maximal in $G$, and so $Phi(G)subseteq H$. Thus, $Z(G)$ is not contained in $Phi(G)$.
More generally, given any nonabelian $p$-group $K$, letting $G=Ktimes C_p$ gives you a group with center $Z(G) = Z(K)times C_p$, and with $K$ maximal and hence $Phi(G)subseteq K$, hence a counterexample to the assertion to $Z(G)subseteq Phi(G)$.
answered 2 days ago
Arturo Magidin
261k32585906
1
Thank you so much.
– Yasmin
2 days ago
add a comment |
Let $H$ be the nonabelian group of order $p^3$ and exponent $p$. Let $C_p$ be a cyclic group of order $p$. Let $G=Htimes C_p$. Then $Z(G)$ is a direct product of two copies of $C_p$, one contained in $H$, the other the second direct factor of $G$.
However, $H$ is maximal in $G$, and so $Phi(G)subseteq H$. Thus, $Z(G)$ is not contained in $Phi(G)$.
More generally, given any nonabelian $p$-group $K$, letting $G=Ktimes C_p$ gives you a group with center $Z(G) = Z(K)times C_p$, and with $K$ maximal and hence $Phi(G)subseteq K$, hence a counterexample to the assertion to $Z(G)subseteq Phi(G)$.
answered 2 days ago
Arturo Magidin
261k32585906
1
Thank you so much.
– Yasmin
2 days ago
add a comment |
Let $H$ be the nonabelian group of order $p^3$ and exponent $p$. Let $C_p$ be a cyclic group of order $p$. Let $G=Htimes C_p$. Then $Z(G)$ is a direct product of two copies of $C_p$, one contained in $H$, the other the second direct factor of $G$.
However, $H$ is maximal in $G$, and so $Phi(G)subseteq H$. Thus, $Z(G)$ is not contained in $Phi(G)$.
More generally, given any nonabelian $p$-group $K$, letting $G=Ktimes C_p$ gives you a group with center $Z(G) = Z(K)times C_p$, and with $K$ maximal and hence $Phi(G)subseteq K$, hence a counterexample to the assertion to $Z(G)subseteq Phi(G)$.
answered 2 days ago
Arturo Magidin
261k32585906
Let $H$ be the nonabelian group of order $p^3$ and exponent $p$. Let $C_p$ be a cyclic group of order $p$. Let $G=Htimes C_p$. Then $Z(G)$ is a direct product of two copies of $C_p$, one contained in $H$, the other the second direct factor of $G$.
However, $H$ is maximal in $G$, and so $Phi(G)subseteq H$. Thus, $Z(G)$ is not contained in $Phi(G)$.
More generally, given any nonabelian $p$-group $K$, letting $G=Ktimes C_p$ gives you a group with center $Z(G) = Z(K)times C_p$, and with $K$ maximal and hence $Phi(G)subseteq K$, hence a counterexample to the assertion to $Z(G)subseteq Phi(G)$.
answered 2 days ago
Arturo Magidin
261k32585906
edited 2 days ago
answered 2 days ago
Arturo Magidin
261k32585906
answered 2 days ago
Arturo Magidin
261k32585906
answered 2 days ago
Arturo Magidin
261k32585906
261k32585906
1
Thank you so much.
– Yasmin
2 days ago
add a comment |
1
Thank you so much.
– Yasmin
2 days ago
1
1
Thank you so much.
– Yasmin
2 days ago
Thank you so much.
– Yasmin
2 days ago
add a comment |
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@Yasmin: Your edit does not address the original problem: why did you think it was true, or why did you want to prove it? As to what you write now, you “ask” whether it is true, and then you say you “understand” it isn’t. That makes the question self-contradictory.
– Arturo Magidin
2 days ago
@ArturoMagidin Would you please suggest how to edit it in a right way?
– Yasmin
2 days ago
@Yasmin: Add context. Add why you were interested in this; why you thought it might be true. Do not falsify the record by writing nonsense after the fact when it turned out that what you wanted to prove is actually false. That would be a good place to start, instead of asking people to do your work for you, again.
– Arturo Magidin
2 days ago
2
The general point is that if you do not know whether something is true or not, then you should not say "help me prove this", you should say "help me decide whether this is true". Otherwise people might waste time trying to prove something that isn't true. You should also say where the problem comes from: is it a homework exercise, is it a problem from a book, is it your own problem? (Occasionally problems from books are wrong, and of course in that case everybody would be confused!)
– Derek Holt
2 days ago
1
I have voted to reopen, because the question itself has some interest and it has been correctly answered.
– Derek Holt
2 days ago