Schur multiplier of direct product
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Let $A$ and $B$ be arbitrary finite groups with $(|A|,|B|)=1$. Let $M(G)$ be the Schur multiplier of the group $G$. Problem 5A.8.(b) in Isaacs' Finite Group Theory asks us to show that $M(A times B) cong M(A) times M(B) $.
We have that $|M(A times B)| ge |M(A)| |M(B)| $ by (a). I anticipate an argument using Schur-Zassenhaus, but I am having trouble applying the coprime condition to the structure of the extension.
finite-groups
asked Nov 21 at 7:58
cas
312
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up vote
1
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Let $A$ and $B$ be arbitrary finite groups with $(|A|,|B|)=1$. Let $M(G)$ be the Schur multiplier of the group $G$. Problem 5A.8.(b) in Isaacs' Finite Group Theory asks us to show that $M(A times B) cong M(A) times M(B) $.
We have that $|M(A times B)| ge |M(A)| |M(B)| $ by (a). I anticipate an argument using Schur-Zassenhaus, but I am having trouble applying the coprime condition to the structure of the extension.
finite-groups
asked Nov 21 at 7:58
cas
312
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $A$ and $B$ be arbitrary finite groups with $(|A|,|B|)=1$. Let $M(G)$ be the Schur multiplier of the group $G$. Problem 5A.8.(b) in Isaacs' Finite Group Theory asks us to show that $M(A times B) cong M(A) times M(B) $.
We have that $|M(A times B)| ge |M(A)| |M(B)| $ by (a). I anticipate an argument using Schur-Zassenhaus, but I am having trouble applying the coprime condition to the structure of the extension.
finite-groups
asked Nov 21 at 7:58
cas
312
Let $A$ and $B$ be arbitrary finite groups with $(|A|,|B|)=1$. Let $M(G)$ be the Schur multiplier of the group $G$. Problem 5A.8.(b) in Isaacs' Finite Group Theory asks us to show that $M(A times B) cong M(A) times M(B) $.
We have that $|M(A times B)| ge |M(A)| |M(B)| $ by (a). I anticipate an argument using Schur-Zassenhaus, but I am having trouble applying the coprime condition to the structure of the extension.
finite-groups
finite-groups
asked Nov 21 at 7:58
cas
312
asked Nov 21 at 7:58
cas
312
asked Nov 21 at 7:58
cas
312
asked Nov 21 at 7:58
cas
312
asked Nov 21 at 7:58
cas
312
312
add a comment |
add a comment |
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