Mixing
The terminology of the group $a^ib^j$ for $0leq i < m$ and $0leq j < n$, i.e. $a^m=1$ and $b^n=1$.
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Suppose I have a group that consists of all elements of type $a^ib^j$ for $0leq i < m$ and $0leq j < n$, i.e. $a^m=1$ and $b^n=1$. Here are my questions:
1) What are $a$ and $b$ called?
2) What are $m$ and $n$ called?
Any other basic related terminology I should know for this case?
abstract-algebra group-theory terminology
edited Jul 26 at 15:56
Shaun
7,31592972
asked Jul 26 at 15:34
Baron Yugovich
1225
 |Â
show 5 more comments
up vote
1
down vote
favorite
Suppose I have a group that consists of all elements of type $a^ib^j$ for $0leq i < m$ and $0leq j < n$, i.e. $a^m=1$ and $b^n=1$. Here are my questions:
1) What are $a$ and $b$ called?
2) What are $m$ and $n$ called?
Any other basic related terminology I should know for this case?
abstract-algebra group-theory terminology
edited Jul 26 at 15:56
Shaun
7,31592972
asked Jul 26 at 15:34
Baron Yugovich
1225
Do you mean "generator" for (1) and "order" for (2)?
– Alan Wang
Jul 26 at 15:37
Not sure, that's why I'm asking :) Does the term generator apply even if there is more than one, like in the case above, a and b?
– Baron Yugovich
Jul 26 at 15:39
Yes. A group can have more than one generator. And in your case, $a,b$ can be said to be a generating set of the group.
– Alan Wang
Jul 26 at 15:41
Thank you so much. Is there a specific terms for this type of group, i.e. is it called cyclic or something?
– Baron Yugovich
Jul 26 at 15:42
"Finitely generated" group
– Alan Wang
Jul 26 at 15:42
 |Â
show 5 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose I have a group that consists of all elements of type $a^ib^j$ for $0leq i < m$ and $0leq j < n$, i.e. $a^m=1$ and $b^n=1$. Here are my questions:
1) What are $a$ and $b$ called?
2) What are $m$ and $n$ called?
Any other basic related terminology I should know for this case?
abstract-algebra group-theory terminology
edited Jul 26 at 15:56
Shaun
7,31592972
asked Jul 26 at 15:34
Baron Yugovich
1225
Suppose I have a group that consists of all elements of type $a^ib^j$ for $0leq i < m$ and $0leq j < n$, i.e. $a^m=1$ and $b^n=1$. Here are my questions:
1) What are $a$ and $b$ called?
2) What are $m$ and $n$ called?
Any other basic related terminology I should know for this case?
abstract-algebra group-theory terminology
edited Jul 26 at 15:56
Shaun
7,31592972
asked Jul 26 at 15:34
Baron Yugovich
1225
edited Jul 26 at 15:56
Shaun
7,31592972
edited Jul 26 at 15:56
Shaun
7,31592972
edited Jul 26 at 15:56
Shaun
7,31592972
7,31592972
asked Jul 26 at 15:34
Baron Yugovich
1225
asked Jul 26 at 15:34
Baron Yugovich
1225
asked Jul 26 at 15:34
Baron Yugovich
1225
1225
Do you mean "generator" for (1) and "order" for (2)?
– Alan Wang
Jul 26 at 15:37
Not sure, that's why I'm asking :) Does the term generator apply even if there is more than one, like in the case above, a and b?
– Baron Yugovich
Jul 26 at 15:39
Yes. A group can have more than one generator. And in your case, $a,b$ can be said to be a generating set of the group.
– Alan Wang
Jul 26 at 15:41
Thank you so much. Is there a specific terms for this type of group, i.e. is it called cyclic or something?
– Baron Yugovich
Jul 26 at 15:42
"Finitely generated" group
– Alan Wang
Jul 26 at 15:42
 |Â
show 5 more comments
Do you mean "generator" for (1) and "order" for (2)?
– Alan Wang
Jul 26 at 15:37
Not sure, that's why I'm asking :) Does the term generator apply even if there is more than one, like in the case above, a and b?
– Baron Yugovich
Jul 26 at 15:39
Yes. A group can have more than one generator. And in your case, $a,b$ can be said to be a generating set of the group.
– Alan Wang
Jul 26 at 15:41
Thank you so much. Is there a specific terms for this type of group, i.e. is it called cyclic or something?
– Baron Yugovich
Jul 26 at 15:42
"Finitely generated" group
– Alan Wang
Jul 26 at 15:42
Do you mean "generator" for (1) and "order" for (2)?
– Alan Wang
Jul 26 at 15:37
Do you mean "generator" for (1) and "order" for (2)?
– Alan Wang
Jul 26 at 15:37
Not sure, that's why I'm asking :) Does the term generator apply even if there is more than one, like in the case above, a and b?
– Baron Yugovich
Jul 26 at 15:39
Not sure, that's why I'm asking :) Does the term generator apply even if there is more than one, like in the case above, a and b?
– Baron Yugovich
Jul 26 at 15:39
Yes. A group can have more than one generator. And in your case, $a,b$ can be said to be a generating set of the group.
– Alan Wang
Jul 26 at 15:41
Yes. A group can have more than one generator. And in your case, $a,b$ can be said to be a generating set of the group.
– Alan Wang
Jul 26 at 15:41
Thank you so much. Is there a specific terms for this type of group, i.e. is it called cyclic or something?
– Baron Yugovich
Jul 26 at 15:42
Thank you so much. Is there a specific terms for this type of group, i.e. is it called cyclic or something?
– Baron Yugovich
Jul 26 at 15:42
"Finitely generated" group
– Alan Wang
Jul 26 at 15:42
"Finitely generated" group
– Alan Wang
Jul 26 at 15:42
 |Â
show 5 more comments
1 Answer
1
active
oldest
votes
up vote
5
down vote
accepted
Here $a$ and $b$ are generators of the group and $m$ and $n$ are the orders of the generators $a$ and $b$, respectively.
Assuming $a$ and $b$ commute, a potential presentation for the group is $$langle a, bmid a^m, b^n, ab=barangle.$$
edited Jul 26 at 16:23
joriki
164k10180328
answered Jul 26 at 15:52
Shaun
7,31592972
Well spotted, @MikeEarnest; thank you! I've edited my answer now.
– Shaun
Jul 26 at 16:04
Let's have a specific example, say for example we are looking at remainders modulo 6, what would be the generators for such a group? I.e. minimal, optimal set of generators. Or how about the case with 10?
– Baron Yugovich
Jul 26 at 16:21
Sorry, I meant under multiplication, not addition.
– Baron Yugovich
Jul 26 at 16:44
Also, speaking of separate group, can you please look at math.stackexchange.com/questions/2863572/…? I was not able to google the exact wording, and for some reason got a downvote already :(
– Baron Yugovich
Jul 26 at 16:45
I suggest you ask that as a new question, @BaronYugovich.
– Shaun
Jul 26 at 16:46
 |Â
show 3 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
Here $a$ and $b$ are generators of the group and $m$ and $n$ are the orders of the generators $a$ and $b$, respectively.
Assuming $a$ and $b$ commute, a potential presentation for the group is $$langle a, bmid a^m, b^n, ab=barangle.$$
edited Jul 26 at 16:23
joriki
164k10180328
answered Jul 26 at 15:52
Shaun
7,31592972
Well spotted, @MikeEarnest; thank you! I've edited my answer now.
– Shaun
Jul 26 at 16:04
Let's have a specific example, say for example we are looking at remainders modulo 6, what would be the generators for such a group? I.e. minimal, optimal set of generators. Or how about the case with 10?
– Baron Yugovich
Jul 26 at 16:21
Sorry, I meant under multiplication, not addition.
– Baron Yugovich
Jul 26 at 16:44
Also, speaking of separate group, can you please look at math.stackexchange.com/questions/2863572/…? I was not able to google the exact wording, and for some reason got a downvote already :(
– Baron Yugovich
Jul 26 at 16:45
I suggest you ask that as a new question, @BaronYugovich.
– Shaun
Jul 26 at 16:46
 |Â
show 3 more comments
up vote
5
down vote
accepted
Here $a$ and $b$ are generators of the group and $m$ and $n$ are the orders of the generators $a$ and $b$, respectively.
Assuming $a$ and $b$ commute, a potential presentation for the group is $$langle a, bmid a^m, b^n, ab=barangle.$$
edited Jul 26 at 16:23
joriki
164k10180328
answered Jul 26 at 15:52
Shaun
7,31592972
Well spotted, @MikeEarnest; thank you! I've edited my answer now.
– Shaun
Jul 26 at 16:04
Let's have a specific example, say for example we are looking at remainders modulo 6, what would be the generators for such a group? I.e. minimal, optimal set of generators. Or how about the case with 10?
– Baron Yugovich
Jul 26 at 16:21
Sorry, I meant under multiplication, not addition.
– Baron Yugovich
Jul 26 at 16:44
Also, speaking of separate group, can you please look at math.stackexchange.com/questions/2863572/…? I was not able to google the exact wording, and for some reason got a downvote already :(
– Baron Yugovich
Jul 26 at 16:45
I suggest you ask that as a new question, @BaronYugovich.
– Shaun
Jul 26 at 16:46
 |Â
show 3 more comments
up vote
5
down vote
accepted
up vote
5
down vote
accepted
Here $a$ and $b$ are generators of the group and $m$ and $n$ are the orders of the generators $a$ and $b$, respectively.
Assuming $a$ and $b$ commute, a potential presentation for the group is $$langle a, bmid a^m, b^n, ab=barangle.$$
edited Jul 26 at 16:23
joriki
164k10180328
answered Jul 26 at 15:52
Shaun
7,31592972
Here $a$ and $b$ are generators of the group and $m$ and $n$ are the orders of the generators $a$ and $b$, respectively.
Assuming $a$ and $b$ commute, a potential presentation for the group is $$langle a, bmid a^m, b^n, ab=barangle.$$
edited Jul 26 at 16:23
joriki
164k10180328
answered Jul 26 at 15:52
Shaun
7,31592972
edited Jul 26 at 16:23
joriki
164k10180328
edited Jul 26 at 16:23
joriki
164k10180328
edited Jul 26 at 16:23
joriki
164k10180328
164k10180328
answered Jul 26 at 15:52
Shaun
7,31592972
answered Jul 26 at 15:52
Shaun
7,31592972
answered Jul 26 at 15:52
Shaun
7,31592972
7,31592972
Well spotted, @MikeEarnest; thank you! I've edited my answer now.
– Shaun
Jul 26 at 16:04
Let's have a specific example, say for example we are looking at remainders modulo 6, what would be the generators for such a group? I.e. minimal, optimal set of generators. Or how about the case with 10?
– Baron Yugovich
Jul 26 at 16:21
Sorry, I meant under multiplication, not addition.
– Baron Yugovich
Jul 26 at 16:44
Also, speaking of separate group, can you please look at math.stackexchange.com/questions/2863572/…? I was not able to google the exact wording, and for some reason got a downvote already :(
– Baron Yugovich
Jul 26 at 16:45
I suggest you ask that as a new question, @BaronYugovich.
– Shaun
Jul 26 at 16:46
 |Â
show 3 more comments
Well spotted, @MikeEarnest; thank you! I've edited my answer now.
– Shaun
Jul 26 at 16:04
Let's have a specific example, say for example we are looking at remainders modulo 6, what would be the generators for such a group? I.e. minimal, optimal set of generators. Or how about the case with 10?
– Baron Yugovich
Jul 26 at 16:21
Sorry, I meant under multiplication, not addition.
– Baron Yugovich
Jul 26 at 16:44
Also, speaking of separate group, can you please look at math.stackexchange.com/questions/2863572/…? I was not able to google the exact wording, and for some reason got a downvote already :(
– Baron Yugovich
Jul 26 at 16:45
I suggest you ask that as a new question, @BaronYugovich.
– Shaun
Jul 26 at 16:46
Well spotted, @MikeEarnest; thank you! I've edited my answer now.
– Shaun
Jul 26 at 16:04
Well spotted, @MikeEarnest; thank you! I've edited my answer now.
– Shaun
Jul 26 at 16:04
Let's have a specific example, say for example we are looking at remainders modulo 6, what would be the generators for such a group? I.e. minimal, optimal set of generators. Or how about the case with 10?
– Baron Yugovich
Jul 26 at 16:21
Let's have a specific example, say for example we are looking at remainders modulo 6, what would be the generators for such a group? I.e. minimal, optimal set of generators. Or how about the case with 10?
– Baron Yugovich
Jul 26 at 16:21
Sorry, I meant under multiplication, not addition.
– Baron Yugovich
Jul 26 at 16:44
Sorry, I meant under multiplication, not addition.
– Baron Yugovich
Jul 26 at 16:44
Also, speaking of separate group, can you please look at math.stackexchange.com/questions/2863572/…? I was not able to google the exact wording, and for some reason got a downvote already :(
– Baron Yugovich
Jul 26 at 16:45
Also, speaking of separate group, can you please look at math.stackexchange.com/questions/2863572/…? I was not able to google the exact wording, and for some reason got a downvote already :(
– Baron Yugovich
Jul 26 at 16:45
I suggest you ask that as a new question, @BaronYugovich.
– Shaun
Jul 26 at 16:46
I suggest you ask that as a new question, @BaronYugovich.
– Shaun
Jul 26 at 16:46
 |Â
show 3 more comments
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Do you mean "generator" for (1) and "order" for (2)?
– Alan Wang
Jul 26 at 15:37
Not sure, that's why I'm asking :) Does the term generator apply even if there is more than one, like in the case above, a and b?
– Baron Yugovich
Jul 26 at 15:39
Yes. A group can have more than one generator. And in your case, $a,b$ can be said to be a generating set of the group.
– Alan Wang
Jul 26 at 15:41
Thank you so much. Is there a specific terms for this type of group, i.e. is it called cyclic or something?
– Baron Yugovich
Jul 26 at 15:42
"Finitely generated" group
– Alan Wang
Jul 26 at 15:42