Presentation of (presumably) inner direct product $G_1times G_2$
$begingroup$
Let $G_1=langle S_1mid R_1rangle, G_2=langle S_2 mid R_2 rangle $ and $S_1cap S_2=emptyset$.
Show that $G_1times G_2=langle S_1cup S_2mid R_1cup R_2cup Rrangle$ where $R={S^{-1}T^{-1}ST=1}, Sin S_1, Tin S_2$.
Question:
Since $S_1cup S_2$ isn't a subgroup of the outer direct product, one should probably identify the outer with the inner product for given presentations. Is that correct? If yes, what are the factors of the inner product that is isomorphic to $G_1times G_2$? I would assume they are the smallest normal subgroups $N_1$ and $N_2$ containing $S_1$ and $S_2$.
abstract-algebra group-theory group-presentation direct-product
edited Dec 3 '18 at 0:54
Shaun
8,893113681
asked Jul 2 '17 at 17:06
user424862user424862
44027
$endgroup$
add a comment |
$begingroup$
Let $G_1=langle S_1mid R_1rangle, G_2=langle S_2 mid R_2 rangle $ and $S_1cap S_2=emptyset$.
Show that $G_1times G_2=langle S_1cup S_2mid R_1cup R_2cup Rrangle$ where $R={S^{-1}T^{-1}ST=1}, Sin S_1, Tin S_2$.
Question:
Since $S_1cup S_2$ isn't a subgroup of the outer direct product, one should probably identify the outer with the inner product for given presentations. Is that correct? If yes, what are the factors of the inner product that is isomorphic to $G_1times G_2$? I would assume they are the smallest normal subgroups $N_1$ and $N_2$ containing $S_1$ and $S_2$.
abstract-algebra group-theory group-presentation direct-product
edited Dec 3 '18 at 0:54
Shaun
8,893113681
asked Jul 2 '17 at 17:06
user424862user424862
44027
$endgroup$
2
$begingroup$
It seems you're talking about presentations of groups and it has nothing to do with representation theory.
$endgroup$
– Orat
Jul 2 '17 at 20:15
$begingroup$
You're right. I edited the tags.
$endgroup$
– user424862
Jul 2 '17 at 20:25
$begingroup$
Fixed all occurrences of "representation" as such.
$endgroup$
– Cameron Williams
Jul 2 '17 at 20:27
$begingroup$
Well, it is easy to see that there's an epimorphism $f:<S_1cup S_2>to G_1times G_2$ and that $R_1cup R_2cup Rsubsetker(f)$. The difficult part is to show that actually the kernel is generated by this set. Not sure yet how to do it.
$endgroup$
– freakish
Jul 2 '17 at 21:04
add a comment |
$begingroup$
Let $G_1=langle S_1mid R_1rangle, G_2=langle S_2 mid R_2 rangle $ and $S_1cap S_2=emptyset$.
Show that $G_1times G_2=langle S_1cup S_2mid R_1cup R_2cup Rrangle$ where $R={S^{-1}T^{-1}ST=1}, Sin S_1, Tin S_2$.
Question:
Since $S_1cup S_2$ isn't a subgroup of the outer direct product, one should probably identify the outer with the inner product for given presentations. Is that correct? If yes, what are the factors of the inner product that is isomorphic to $G_1times G_2$? I would assume they are the smallest normal subgroups $N_1$ and $N_2$ containing $S_1$ and $S_2$.
abstract-algebra group-theory group-presentation direct-product
edited Dec 3 '18 at 0:54
Shaun
8,893113681
asked Jul 2 '17 at 17:06
user424862user424862
44027
$endgroup$
Let $G_1=langle S_1mid R_1rangle, G_2=langle S_2 mid R_2 rangle $ and $S_1cap S_2=emptyset$.
Show that $G_1times G_2=langle S_1cup S_2mid R_1cup R_2cup Rrangle$ where $R={S^{-1}T^{-1}ST=1}, Sin S_1, Tin S_2$.
Question:
Since $S_1cup S_2$ isn't a subgroup of the outer direct product, one should probably identify the outer with the inner product for given presentations. Is that correct? If yes, what are the factors of the inner product that is isomorphic to $G_1times G_2$? I would assume they are the smallest normal subgroups $N_1$ and $N_2$ containing $S_1$ and $S_2$.
abstract-algebra group-theory group-presentation direct-product
abstract-algebra group-theory group-presentation direct-product
edited Dec 3 '18 at 0:54
Shaun
8,893113681
asked Jul 2 '17 at 17:06
user424862user424862
44027
edited Dec 3 '18 at 0:54
Shaun
8,893113681
asked Jul 2 '17 at 17:06
user424862user424862
44027
edited Dec 3 '18 at 0:54
Shaun
8,893113681
edited Dec 3 '18 at 0:54
Shaun
8,893113681
edited Dec 3 '18 at 0:54
Shaun
8,893113681
8,893113681
asked Jul 2 '17 at 17:06
user424862user424862
44027
asked Jul 2 '17 at 17:06
user424862user424862
44027
asked Jul 2 '17 at 17:06
user424862user424862
44027
44027
2
$begingroup$
It seems you're talking about presentations of groups and it has nothing to do with representation theory.
$endgroup$
– Orat
Jul 2 '17 at 20:15
$begingroup$
You're right. I edited the tags.
$endgroup$
– user424862
Jul 2 '17 at 20:25
$begingroup$
Fixed all occurrences of "representation" as such.
$endgroup$
– Cameron Williams
Jul 2 '17 at 20:27
$begingroup$
Well, it is easy to see that there's an epimorphism $f:<S_1cup S_2>to G_1times G_2$ and that $R_1cup R_2cup Rsubsetker(f)$. The difficult part is to show that actually the kernel is generated by this set. Not sure yet how to do it.
$endgroup$
– freakish
Jul 2 '17 at 21:04
add a comment |
2
$begingroup$
It seems you're talking about presentations of groups and it has nothing to do with representation theory.
$endgroup$
– Orat
Jul 2 '17 at 20:15
$begingroup$
You're right. I edited the tags.
$endgroup$
– user424862
Jul 2 '17 at 20:25
$begingroup$
Fixed all occurrences of "representation" as such.
$endgroup$
– Cameron Williams
Jul 2 '17 at 20:27
$begingroup$
Well, it is easy to see that there's an epimorphism $f:<S_1cup S_2>to G_1times G_2$ and that $R_1cup R_2cup Rsubsetker(f)$. The difficult part is to show that actually the kernel is generated by this set. Not sure yet how to do it.
$endgroup$
– freakish
Jul 2 '17 at 21:04
2
2
$begingroup$
It seems you're talking about presentations of groups and it has nothing to do with representation theory.
$endgroup$
– Orat
Jul 2 '17 at 20:15
$begingroup$
It seems you're talking about presentations of groups and it has nothing to do with representation theory.
$endgroup$
– Orat
Jul 2 '17 at 20:15
$begingroup$
You're right. I edited the tags.
$endgroup$
– user424862
Jul 2 '17 at 20:25
$begingroup$
You're right. I edited the tags.
$endgroup$
– user424862
Jul 2 '17 at 20:25
$begingroup$
Fixed all occurrences of "representation" as such.
$endgroup$
– Cameron Williams
Jul 2 '17 at 20:27
$begingroup$
Fixed all occurrences of "representation" as such.
$endgroup$
– Cameron Williams
Jul 2 '17 at 20:27
$begingroup$
Well, it is easy to see that there's an epimorphism $f:<S_1cup S_2>to G_1times G_2$ and that $R_1cup R_2cup Rsubsetker(f)$. The difficult part is to show that actually the kernel is generated by this set. Not sure yet how to do it.
$endgroup$
– freakish
Jul 2 '17 at 21:04
$begingroup$
Well, it is easy to see that there's an epimorphism $f:<S_1cup S_2>to G_1times G_2$ and that $R_1cup R_2cup Rsubsetker(f)$. The difficult part is to show that actually the kernel is generated by this set. Not sure yet how to do it.
$endgroup$
– freakish
Jul 2 '17 at 21:04
add a comment |
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2
$begingroup$
It seems you're talking about presentations of groups and it has nothing to do with representation theory.
$endgroup$
– Orat
Jul 2 '17 at 20:15
$begingroup$
You're right. I edited the tags.
$endgroup$
– user424862
Jul 2 '17 at 20:25
$begingroup$
Fixed all occurrences of "representation" as such.
$endgroup$
– Cameron Williams
Jul 2 '17 at 20:27
$begingroup$
Well, it is easy to see that there's an epimorphism $f:<S_1cup S_2>to G_1times G_2$ and that $R_1cup R_2cup Rsubsetker(f)$. The difficult part is to show that actually the kernel is generated by this set. Not sure yet how to do it.
$endgroup$
– freakish
Jul 2 '17 at 21:04