Basic idea of generators of a group
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Started studying group theory and I couldn't understand the following:
Given group $G = langle a,b rangle$ with binary operation marked $*$ where $a,b$ are the generators of $G$ I understand that $a,a*a,a*b,b*ain G$
but does it mean that also $a*b*ain G$?
I think that it's true.
group-theory
edited Nov 24 at 21:59
the_fox
2,2941430
asked Nov 18 at 17:29
Yaron Scherf
1187
add a comment |
up vote
0
down vote
favorite
Started studying group theory and I couldn't understand the following:
Given group $G = langle a,b rangle$ with binary operation marked $*$ where $a,b$ are the generators of $G$ I understand that $a,a*a,a*b,b*ain G$
but does it mean that also $a*b*ain G$?
I think that it's true.
group-theory
edited Nov 24 at 21:59
the_fox
2,2941430
asked Nov 18 at 17:29
Yaron Scherf
1187
Yes, it is the minimal group which contains both $a$ and $b$. If $G$ is a group and $a,bin G$ then $abin G$ and $abain G$.
– Yanko
Nov 18 at 17:33
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Started studying group theory and I couldn't understand the following:
Given group $G = langle a,b rangle$ with binary operation marked $*$ where $a,b$ are the generators of $G$ I understand that $a,a*a,a*b,b*ain G$
but does it mean that also $a*b*ain G$?
I think that it's true.
group-theory
edited Nov 24 at 21:59
the_fox
2,2941430
asked Nov 18 at 17:29
Yaron Scherf
1187
Started studying group theory and I couldn't understand the following:
Given group $G = langle a,b rangle$ with binary operation marked $*$ where $a,b$ are the generators of $G$ I understand that $a,a*a,a*b,b*ain G$
but does it mean that also $a*b*ain G$?
I think that it's true.
group-theory
group-theory
edited Nov 24 at 21:59
the_fox
2,2941430
asked Nov 18 at 17:29
Yaron Scherf
1187
edited Nov 24 at 21:59
the_fox
2,2941430
asked Nov 18 at 17:29
Yaron Scherf
1187
edited Nov 24 at 21:59
the_fox
2,2941430
edited Nov 24 at 21:59
the_fox
2,2941430
edited Nov 24 at 21:59
the_fox
2,2941430
2,2941430
asked Nov 18 at 17:29
Yaron Scherf
1187
asked Nov 18 at 17:29
Yaron Scherf
1187
asked Nov 18 at 17:29
Yaron Scherf
1187
1187
Yes, it is the minimal group which contains both $a$ and $b$. If $G$ is a group and $a,bin G$ then $abin G$ and $abain G$.
– Yanko
Nov 18 at 17:33
add a comment |
Yes, it is the minimal group which contains both $a$ and $b$. If $G$ is a group and $a,bin G$ then $abin G$ and $abain G$.
– Yanko
Nov 18 at 17:33
Yes, it is the minimal group which contains both $a$ and $b$. If $G$ is a group and $a,bin G$ then $abin G$ and $abain G$.
– Yanko
Nov 18 at 17:33
Yes, it is the minimal group which contains both $a$ and $b$. If $G$ is a group and $a,bin G$ then $abin G$ and $abain G$.
– Yanko
Nov 18 at 17:33
add a comment |
1 Answer
1
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votes
up vote
4
down vote
accepted
Yes, it does; it means all finite products of $a$, $b$, and their inverses, all with respect to $*$, are in $G$ (and, in fact, there are no other elements in $G$).
answered Nov 18 at 17:33
Shaun
8,100113577
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
Yes, it does; it means all finite products of $a$, $b$, and their inverses, all with respect to $*$, are in $G$ (and, in fact, there are no other elements in $G$).
answered Nov 18 at 17:33
Shaun
8,100113577
add a comment |
up vote
4
down vote
accepted
Yes, it does; it means all finite products of $a$, $b$, and their inverses, all with respect to $*$, are in $G$ (and, in fact, there are no other elements in $G$).
answered Nov 18 at 17:33
Shaun
8,100113577
add a comment |
up vote
4
down vote
accepted
up vote
4
down vote
accepted
Yes, it does; it means all finite products of $a$, $b$, and their inverses, all with respect to $*$, are in $G$ (and, in fact, there are no other elements in $G$).
answered Nov 18 at 17:33
Shaun
8,100113577
Yes, it does; it means all finite products of $a$, $b$, and their inverses, all with respect to $*$, are in $G$ (and, in fact, there are no other elements in $G$).
answered Nov 18 at 17:33
Shaun
8,100113577
edited Nov 24 at 20:59
answered Nov 18 at 17:33
Shaun
8,100113577
answered Nov 18 at 17:33
Shaun
8,100113577
answered Nov 18 at 17:33
Shaun
8,100113577
8,100113577
add a comment |
add a comment |
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Yes, it is the minimal group which contains both $a$ and $b$. If $G$ is a group and $a,bin G$ then $abin G$ and $abain G$.
– Yanko
Nov 18 at 17:33