Proving a direct product of groups is a group
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I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance.
Let $n geq 1$ be an integer. Suppose ${G_1,...,G_n}$ is a finite set of groups. Define the set $G = G_1 times G_2 times ... times G_n$ of ordered n-tuples $(g_1,g_2,...,g_n)$ where $g_i in G_i$. Show that the operation $bullet$: $G times G rightarrow G$ given by $(g_1,g_2,...,g_n) bullet (g'_1,g'_2,...,g'_n) = (g_1g'_1,g_2g'_2,...,g_ng'_n)$ makes $G$ into a group.
Proof of associativity: $((g_1,g_2,...,g_n) bullet (g'_1,g'_2,...g'_n)) bullet (g''_1,g''_2,...,g''_n)$ = $(g_1g'_1,g_2g'_2,...,g_ng'_n) bullet (g''_1,g''_2,...,g''_n) $ = $(g_1g'_1g''_1,g_2g'_2g''_2,...,g_ng'_ng''_n)$ = $(g_1,g_2,...,g_n) bullet (g'_1g''_1,g'_2g''_2,...,g'_ng''_n) = (g_1,g_2,...,g_n) bullet ((g'_1,g'_2,...,g'_n) bullet (g''_1,g''_2,...,g''_n))$
Proof of the existence of an identity For each $G_i in G$, take its identity element $e_i$ and multiply it as follows: $(g_1,g_2,...,g_n) bullet (e_1,e_2,...,e_3)$ = $(g_1,g_2,...,g_n)$.
Proof of the existence of an inverse: Since each $G_i in G$ is a group, simply takes its multiplicative inverse $g_i^{-1}$ and multiply it: $(g_1,g_2,...,g_n) bullet (g_1^{-1},g_2^{-1},...,g_n^{-1}) = (e_1,e_2,...,e_n)$.
proof-verification finite-groups direct-product
edited Dec 22 '18 at 4:09
Shaun
9,241113684
asked Feb 24 '15 at 6:07
letsmakemuffinstogetherletsmakemuffinstogether
9441717
$endgroup$
add a comment |
$begingroup$
I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance.
Let $n geq 1$ be an integer. Suppose ${G_1,...,G_n}$ is a finite set of groups. Define the set $G = G_1 times G_2 times ... times G_n$ of ordered n-tuples $(g_1,g_2,...,g_n)$ where $g_i in G_i$. Show that the operation $bullet$: $G times G rightarrow G$ given by $(g_1,g_2,...,g_n) bullet (g'_1,g'_2,...,g'_n) = (g_1g'_1,g_2g'_2,...,g_ng'_n)$ makes $G$ into a group.
Proof of associativity: $((g_1,g_2,...,g_n) bullet (g'_1,g'_2,...g'_n)) bullet (g''_1,g''_2,...,g''_n)$ = $(g_1g'_1,g_2g'_2,...,g_ng'_n) bullet (g''_1,g''_2,...,g''_n) $ = $(g_1g'_1g''_1,g_2g'_2g''_2,...,g_ng'_ng''_n)$ = $(g_1,g_2,...,g_n) bullet (g'_1g''_1,g'_2g''_2,...,g'_ng''_n) = (g_1,g_2,...,g_n) bullet ((g'_1,g'_2,...,g'_n) bullet (g''_1,g''_2,...,g''_n))$
Proof of the existence of an identity For each $G_i in G$, take its identity element $e_i$ and multiply it as follows: $(g_1,g_2,...,g_n) bullet (e_1,e_2,...,e_3)$ = $(g_1,g_2,...,g_n)$.
Proof of the existence of an inverse: Since each $G_i in G$ is a group, simply takes its multiplicative inverse $g_i^{-1}$ and multiply it: $(g_1,g_2,...,g_n) bullet (g_1^{-1},g_2^{-1},...,g_n^{-1}) = (e_1,e_2,...,e_n)$.
proof-verification finite-groups direct-product
edited Dec 22 '18 at 4:09
Shaun
9,241113684
asked Feb 24 '15 at 6:07
letsmakemuffinstogetherletsmakemuffinstogether
9441717
$endgroup$
1
$begingroup$
Looks fine, and extremely boring as this thing usually is. Just remember to briefly explain that in the associativity part you're using the corresponding associativity of $;G_i;$ in each coordinate, and don't drop the parentheses: $$left((g_1,...)(g_1',...)right)cdot(g_1'',...)=((g_1g_1')g_1'',...)=(g_1 (g_1'cdot g_1''),...);,;;etc.$$
$endgroup$
– Timbuc
Feb 24 '15 at 6:27
$begingroup$
Thank you very much Timbuc for that tip on associativity.
$endgroup$
– letsmakemuffinstogether
Feb 25 '15 at 3:28
add a comment |
$begingroup$
I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance.
Let $n geq 1$ be an integer. Suppose ${G_1,...,G_n}$ is a finite set of groups. Define the set $G = G_1 times G_2 times ... times G_n$ of ordered n-tuples $(g_1,g_2,...,g_n)$ where $g_i in G_i$. Show that the operation $bullet$: $G times G rightarrow G$ given by $(g_1,g_2,...,g_n) bullet (g'_1,g'_2,...,g'_n) = (g_1g'_1,g_2g'_2,...,g_ng'_n)$ makes $G$ into a group.
Proof of associativity: $((g_1,g_2,...,g_n) bullet (g'_1,g'_2,...g'_n)) bullet (g''_1,g''_2,...,g''_n)$ = $(g_1g'_1,g_2g'_2,...,g_ng'_n) bullet (g''_1,g''_2,...,g''_n) $ = $(g_1g'_1g''_1,g_2g'_2g''_2,...,g_ng'_ng''_n)$ = $(g_1,g_2,...,g_n) bullet (g'_1g''_1,g'_2g''_2,...,g'_ng''_n) = (g_1,g_2,...,g_n) bullet ((g'_1,g'_2,...,g'_n) bullet (g''_1,g''_2,...,g''_n))$
Proof of the existence of an identity For each $G_i in G$, take its identity element $e_i$ and multiply it as follows: $(g_1,g_2,...,g_n) bullet (e_1,e_2,...,e_3)$ = $(g_1,g_2,...,g_n)$.
Proof of the existence of an inverse: Since each $G_i in G$ is a group, simply takes its multiplicative inverse $g_i^{-1}$ and multiply it: $(g_1,g_2,...,g_n) bullet (g_1^{-1},g_2^{-1},...,g_n^{-1}) = (e_1,e_2,...,e_n)$.
proof-verification finite-groups direct-product
edited Dec 22 '18 at 4:09
Shaun
9,241113684
asked Feb 24 '15 at 6:07
letsmakemuffinstogetherletsmakemuffinstogether
9441717
$endgroup$
I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance.
Let $n geq 1$ be an integer. Suppose ${G_1,...,G_n}$ is a finite set of groups. Define the set $G = G_1 times G_2 times ... times G_n$ of ordered n-tuples $(g_1,g_2,...,g_n)$ where $g_i in G_i$. Show that the operation $bullet$: $G times G rightarrow G$ given by $(g_1,g_2,...,g_n) bullet (g'_1,g'_2,...,g'_n) = (g_1g'_1,g_2g'_2,...,g_ng'_n)$ makes $G$ into a group.
Proof of associativity: $((g_1,g_2,...,g_n) bullet (g'_1,g'_2,...g'_n)) bullet (g''_1,g''_2,...,g''_n)$ = $(g_1g'_1,g_2g'_2,...,g_ng'_n) bullet (g''_1,g''_2,...,g''_n) $ = $(g_1g'_1g''_1,g_2g'_2g''_2,...,g_ng'_ng''_n)$ = $(g_1,g_2,...,g_n) bullet (g'_1g''_1,g'_2g''_2,...,g'_ng''_n) = (g_1,g_2,...,g_n) bullet ((g'_1,g'_2,...,g'_n) bullet (g''_1,g''_2,...,g''_n))$
Proof of the existence of an identity For each $G_i in G$, take its identity element $e_i$ and multiply it as follows: $(g_1,g_2,...,g_n) bullet (e_1,e_2,...,e_3)$ = $(g_1,g_2,...,g_n)$.
Proof of the existence of an inverse: Since each $G_i in G$ is a group, simply takes its multiplicative inverse $g_i^{-1}$ and multiply it: $(g_1,g_2,...,g_n) bullet (g_1^{-1},g_2^{-1},...,g_n^{-1}) = (e_1,e_2,...,e_n)$.
proof-verification finite-groups direct-product
proof-verification finite-groups direct-product
edited Dec 22 '18 at 4:09
Shaun
9,241113684
asked Feb 24 '15 at 6:07
letsmakemuffinstogetherletsmakemuffinstogether
9441717
edited Dec 22 '18 at 4:09
Shaun
9,241113684
asked Feb 24 '15 at 6:07
letsmakemuffinstogetherletsmakemuffinstogether
9441717
edited Dec 22 '18 at 4:09
Shaun
9,241113684
edited Dec 22 '18 at 4:09
Shaun
9,241113684
edited Dec 22 '18 at 4:09
Shaun
9,241113684
9,241113684
asked Feb 24 '15 at 6:07
letsmakemuffinstogetherletsmakemuffinstogether
9441717
asked Feb 24 '15 at 6:07
letsmakemuffinstogetherletsmakemuffinstogether
9441717
asked Feb 24 '15 at 6:07
letsmakemuffinstogetherletsmakemuffinstogether
9441717
9441717
1
$begingroup$
Looks fine, and extremely boring as this thing usually is. Just remember to briefly explain that in the associativity part you're using the corresponding associativity of $;G_i;$ in each coordinate, and don't drop the parentheses: $$left((g_1,...)(g_1',...)right)cdot(g_1'',...)=((g_1g_1')g_1'',...)=(g_1 (g_1'cdot g_1''),...);,;;etc.$$
$endgroup$
– Timbuc
Feb 24 '15 at 6:27
$begingroup$
Thank you very much Timbuc for that tip on associativity.
$endgroup$
– letsmakemuffinstogether
Feb 25 '15 at 3:28
add a comment |
1
$begingroup$
Looks fine, and extremely boring as this thing usually is. Just remember to briefly explain that in the associativity part you're using the corresponding associativity of $;G_i;$ in each coordinate, and don't drop the parentheses: $$left((g_1,...)(g_1',...)right)cdot(g_1'',...)=((g_1g_1')g_1'',...)=(g_1 (g_1'cdot g_1''),...);,;;etc.$$
$endgroup$
– Timbuc
Feb 24 '15 at 6:27
$begingroup$
Thank you very much Timbuc for that tip on associativity.
$endgroup$
– letsmakemuffinstogether
Feb 25 '15 at 3:28
1
1
$begingroup$
Looks fine, and extremely boring as this thing usually is. Just remember to briefly explain that in the associativity part you're using the corresponding associativity of $;G_i;$ in each coordinate, and don't drop the parentheses: $$left((g_1,...)(g_1',...)right)cdot(g_1'',...)=((g_1g_1')g_1'',...)=(g_1 (g_1'cdot g_1''),...);,;;etc.$$
$endgroup$
– Timbuc
Feb 24 '15 at 6:27
$begingroup$
Looks fine, and extremely boring as this thing usually is. Just remember to briefly explain that in the associativity part you're using the corresponding associativity of $;G_i;$ in each coordinate, and don't drop the parentheses: $$left((g_1,...)(g_1',...)right)cdot(g_1'',...)=((g_1g_1')g_1'',...)=(g_1 (g_1'cdot g_1''),...);,;;etc.$$
$endgroup$
– Timbuc
Feb 24 '15 at 6:27
$begingroup$
Thank you very much Timbuc for that tip on associativity.
$endgroup$
– letsmakemuffinstogether
Feb 25 '15 at 3:28
$begingroup$
Thank you very much Timbuc for that tip on associativity.
$endgroup$
– letsmakemuffinstogether
Feb 25 '15 at 3:28
add a comment |
1 Answer
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Just as @Timbuc says, it looks fine (but one could argue that you need to include some reference to the fact that $G$ is closed under $bullet$, but that follows by construction of $bullet$).
answered Dec 22 '18 at 4:07
ShaunShaun
9,241113684
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add a comment |
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$begingroup$
Just as @Timbuc says, it looks fine (but one could argue that you need to include some reference to the fact that $G$ is closed under $bullet$, but that follows by construction of $bullet$).
answered Dec 22 '18 at 4:07
ShaunShaun
9,241113684
$endgroup$
add a comment |
$begingroup$
Just as @Timbuc says, it looks fine (but one could argue that you need to include some reference to the fact that $G$ is closed under $bullet$, but that follows by construction of $bullet$).
answered Dec 22 '18 at 4:07
ShaunShaun
9,241113684
$endgroup$
add a comment |
$begingroup$
Just as @Timbuc says, it looks fine (but one could argue that you need to include some reference to the fact that $G$ is closed under $bullet$, but that follows by construction of $bullet$).
answered Dec 22 '18 at 4:07
ShaunShaun
9,241113684
$endgroup$
Just as @Timbuc says, it looks fine (but one could argue that you need to include some reference to the fact that $G$ is closed under $bullet$, but that follows by construction of $bullet$).
answered Dec 22 '18 at 4:07
ShaunShaun
9,241113684
answered Dec 22 '18 at 4:07
ShaunShaun
9,241113684
answered Dec 22 '18 at 4:07
ShaunShaun
9,241113684
answered Dec 22 '18 at 4:07
ShaunShaun
9,241113684
9,241113684
add a comment |
add a comment |
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1
$begingroup$
Looks fine, and extremely boring as this thing usually is. Just remember to briefly explain that in the associativity part you're using the corresponding associativity of $;G_i;$ in each coordinate, and don't drop the parentheses: $$left((g_1,...)(g_1',...)right)cdot(g_1'',...)=((g_1g_1')g_1'',...)=(g_1 (g_1'cdot g_1''),...);,;;etc.$$
$endgroup$
– Timbuc
Feb 24 '15 at 6:27
$begingroup$
Thank you very much Timbuc for that tip on associativity.
$endgroup$
– letsmakemuffinstogether
Feb 25 '15 at 3:28