A finite group is isomorphic to the direct product of two normal subsets with trivial intersectionUnderstanding the internal direct product of a group.Let N1 and N2 are normal subgroups in the finite group G. Is it true that if N1≃N1 then G∖N1≃G∖N2.?Normal subgroups of direct productProve that $G$ is the internal direct product of their normal subgroups $N_1,N_2,ldots ,N_n$?If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$Proving that product of two quotients = a certain quotient groupFaithful irreducible character of a group with exactly two minimal normal subgroupsFrattini subgroup of a finite elementary abelian $p$-group is trivialIntersection of Frattini subgroup and center of a finite $p$-groupMinimal normal, maximal and isomorphic
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A finite group is isomorphic to the direct product of two normal subsets with trivial intersection
Understanding the internal direct product of a group.Let N1 and N2 are normal subgroups in the finite group G. Is it true that if N1≃N1 then G∖N1≃G∖N2.?Normal subgroups of direct productProve that $G$ is the internal direct product of their normal subgroups $N_1,N_2,ldots ,N_n$?If $H$ is a normal subgroup of a finite group $G$ and $|H|=p^k$ for some prime $p$. show that $H$ is contained in every sylow $p$ subgroup of $G$Proving that product of two quotients = a certain quotient groupFaithful irreducible character of a group with exactly two minimal normal subgroupsFrattini subgroup of a finite elementary abelian $p$-group is trivialIntersection of Frattini subgroup and center of a finite $p$-groupMinimal normal, maximal and isomorphic
$begingroup$
I have to show the following:
Let $G$ be a finite group and $N_1 , N_2$ normal in $G$.
Then $Gcong N_1times N_2$ if and only if $N_1cap N_2 = e$.
I have no idea on either direction, so I would be grateful for any little hint!
Thank you!
group-theory finite-groups
asked May 4 at 9:35
TwoStonesTwoStones
676
$endgroup$
add a comment |
$begingroup$
I have to show the following:
Let $G$ be a finite group and $N_1 , N_2$ normal in $G$.
Then $Gcong N_1times N_2$ if and only if $N_1cap N_2 = e$.
I have no idea on either direction, so I would be grateful for any little hint!
Thank you!
group-theory finite-groups
asked May 4 at 9:35
TwoStonesTwoStones
676
$endgroup$
add a comment |
$begingroup$
I have to show the following:
Let $G$ be a finite group and $N_1 , N_2$ normal in $G$.
Then $Gcong N_1times N_2$ if and only if $N_1cap N_2 = e$.
I have no idea on either direction, so I would be grateful for any little hint!
Thank you!
group-theory finite-groups
asked May 4 at 9:35
TwoStonesTwoStones
676
$endgroup$
I have to show the following:
Let $G$ be a finite group and $N_1 , N_2$ normal in $G$.
Then $Gcong N_1times N_2$ if and only if $N_1cap N_2 = e$.
I have no idea on either direction, so I would be grateful for any little hint!
Thank you!
group-theory finite-groups
group-theory finite-groups
asked May 4 at 9:35
TwoStonesTwoStones
676
asked May 4 at 9:35
TwoStonesTwoStones
676
asked May 4 at 9:35
TwoStonesTwoStones
676
asked May 4 at 9:35
TwoStonesTwoStones
676
asked May 4 at 9:35
TwoStonesTwoStones
676
676
add a comment |
add a comment |
1 Answer
1
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$begingroup$
This is not true unless $N_1,N_2$ generates $G$, in this case let $p_i:Grightarrow G/N_i$ the quotient map, show that $p_1$ iduces an isomorphism $N_2rightarrow G/N_1$. Consider $f:Grightarrow N_1times N_2$ defined $f(x)=(p_1(x),p_2(x))$ show that it is an isomorphism.
answered May 4 at 9:49
Tsemo AristideTsemo Aristide
62k11447
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– TwoStones
May 4 at 10:17
add a comment |
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1 Answer
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1 Answer
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$begingroup$
This is not true unless $N_1,N_2$ generates $G$, in this case let $p_i:Grightarrow G/N_i$ the quotient map, show that $p_1$ iduces an isomorphism $N_2rightarrow G/N_1$. Consider $f:Grightarrow N_1times N_2$ defined $f(x)=(p_1(x),p_2(x))$ show that it is an isomorphism.
answered May 4 at 9:49
Tsemo AristideTsemo Aristide
62k11447
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– TwoStones
May 4 at 10:17
add a comment |
$begingroup$
This is not true unless $N_1,N_2$ generates $G$, in this case let $p_i:Grightarrow G/N_i$ the quotient map, show that $p_1$ iduces an isomorphism $N_2rightarrow G/N_1$. Consider $f:Grightarrow N_1times N_2$ defined $f(x)=(p_1(x),p_2(x))$ show that it is an isomorphism.
answered May 4 at 9:49
Tsemo AristideTsemo Aristide
62k11447
$endgroup$
$begingroup$
Thank you very much!
$endgroup$
– TwoStones
May 4 at 10:17
add a comment |
$begingroup$
This is not true unless $N_1,N_2$ generates $G$, in this case let $p_i:Grightarrow G/N_i$ the quotient map, show that $p_1$ iduces an isomorphism $N_2rightarrow G/N_1$. Consider $f:Grightarrow N_1times N_2$ defined $f(x)=(p_1(x),p_2(x))$ show that it is an isomorphism.
answered May 4 at 9:49
Tsemo AristideTsemo Aristide
62k11447
$endgroup$
This is not true unless $N_1,N_2$ generates $G$, in this case let $p_i:Grightarrow G/N_i$ the quotient map, show that $p_1$ iduces an isomorphism $N_2rightarrow G/N_1$. Consider $f:Grightarrow N_1times N_2$ defined $f(x)=(p_1(x),p_2(x))$ show that it is an isomorphism.
answered May 4 at 9:49
Tsemo AristideTsemo Aristide
62k11447
answered May 4 at 9:49
Tsemo AristideTsemo Aristide
62k11447
answered May 4 at 9:49
Tsemo AristideTsemo Aristide
62k11447
answered May 4 at 9:49
Tsemo AristideTsemo Aristide
62k11447
62k11447
$begingroup$
Thank you very much!
$endgroup$
– TwoStones
May 4 at 10:17
add a comment |
$begingroup$
Thank you very much!
$endgroup$
– TwoStones
May 4 at 10:17
$begingroup$
Thank you very much!
$endgroup$
– TwoStones
May 4 at 10:17
$begingroup$
Thank you very much!
$endgroup$
– TwoStones
May 4 at 10:17
add a comment |
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