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Let $G$ be a group from order $2^n$ defined by: $G=langle a,b: a^2^n-2=b^2=(ab)^2rangle$ .
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Let $G$ be a group from order $2^n$ defined by: $G=langle a,b: a^2^n-2=b^2=(ab)^2rangle$.
find all the subgroups of $G$.
I know that the order of the subgroups of G dividing the order of G so I found $e,b^2, e,b,b^2,b^3 and langle arangle=e,a,a^2,dots, a^2^n-2$.
$(a^ib)^2=a^iba^ib=a^iba^ib^-1b^2=b^2$ . So there is a normal subgroup ⟨a⟩ of order 2n−1, and the square of every element outside of this subgroup is equal to $b^2=a^2^n−2$. So all elements outside of that subgroup have order 4. so thet form subgroups $langle a^ibrangle=e,a^ib,b^2, b^2a^ib$ right?
$(a^ib)^3=a^iba^iba^ib=a^iba^ib^-1b^2a^ib^-1b^2=b^2a^ib^-1b^2=b^2a^ib$
how I preceed?
and do you think about more subgroups? please if you do give me a lead and I will work on it
group-theory finite-groups
asked 2 days ago
Rimon
33
 |Â
show 6 more comments
up vote
-3
down vote
favorite
Let $G$ be a group from order $2^n$ defined by: $G=langle a,b: a^2^n-2=b^2=(ab)^2rangle$.
find all the subgroups of $G$.
I know that the order of the subgroups of G dividing the order of G so I found $e,b^2, e,b,b^2,b^3 and langle arangle=e,a,a^2,dots, a^2^n-2$.
$(a^ib)^2=a^iba^ib=a^iba^ib^-1b^2=b^2$ . So there is a normal subgroup ⟨a⟩ of order 2n−1, and the square of every element outside of this subgroup is equal to $b^2=a^2^n−2$. So all elements outside of that subgroup have order 4. so thet form subgroups $langle a^ibrangle=e,a^ib,b^2, b^2a^ib$ right?
$(a^ib)^3=a^iba^iba^ib=a^iba^ib^-1b^2a^ib^-1b^2=b^2a^ib^-1b^2=b^2a^ib$
how I preceed?
and do you think about more subgroups? please if you do give me a lead and I will work on it
group-theory finite-groups
asked 2 days ago
Rimon
33
Are you doing some assignment or homework regarding to the group presentation given?
– Alan Wang
2 days ago
pressntation but I don't know how to find all the subgroups as needed
– Rimon
2 days ago
2
This question was asked very recently. In fact, unless I'm mistaken this is the third time I see this question within a week. You do know that you should not repost the same question over and over again?
– Jyrki Lahtonen
2 days ago
2
You posted this. You are supposed to EDIT that first version if you have something to add to it. Reposting is simply rude, because it creates orphaned threads. It is also pointless, because anyone into group theory questions will see the version from yesterday. Why do you want to attract negative attention?
– Jyrki Lahtonen
2 days ago
ok I will edit the previous one
– Rimon
2 days ago
 |Â
show 6 more comments
up vote
-3
down vote
favorite
up vote
-3
down vote
favorite
Let $G$ be a group from order $2^n$ defined by: $G=langle a,b: a^2^n-2=b^2=(ab)^2rangle$.
find all the subgroups of $G$.
I know that the order of the subgroups of G dividing the order of G so I found $e,b^2, e,b,b^2,b^3 and langle arangle=e,a,a^2,dots, a^2^n-2$.
$(a^ib)^2=a^iba^ib=a^iba^ib^-1b^2=b^2$ . So there is a normal subgroup ⟨a⟩ of order 2n−1, and the square of every element outside of this subgroup is equal to $b^2=a^2^n−2$. So all elements outside of that subgroup have order 4. so thet form subgroups $langle a^ibrangle=e,a^ib,b^2, b^2a^ib$ right?
$(a^ib)^3=a^iba^iba^ib=a^iba^ib^-1b^2a^ib^-1b^2=b^2a^ib^-1b^2=b^2a^ib$
how I preceed?
and do you think about more subgroups? please if you do give me a lead and I will work on it
group-theory finite-groups
asked 2 days ago
Rimon
33
Let $G$ be a group from order $2^n$ defined by: $G=langle a,b: a^2^n-2=b^2=(ab)^2rangle$.
find all the subgroups of $G$.
I know that the order of the subgroups of G dividing the order of G so I found $e,b^2, e,b,b^2,b^3 and langle arangle=e,a,a^2,dots, a^2^n-2$.
$(a^ib)^2=a^iba^ib=a^iba^ib^-1b^2=b^2$ . So there is a normal subgroup ⟨a⟩ of order 2n−1, and the square of every element outside of this subgroup is equal to $b^2=a^2^n−2$. So all elements outside of that subgroup have order 4. so thet form subgroups $langle a^ibrangle=e,a^ib,b^2, b^2a^ib$ right?
$(a^ib)^3=a^iba^iba^ib=a^iba^ib^-1b^2a^ib^-1b^2=b^2a^ib^-1b^2=b^2a^ib$
how I preceed?
and do you think about more subgroups? please if you do give me a lead and I will work on it
group-theory finite-groups
asked 2 days ago
Rimon
33
edited 18 hours ago
asked 2 days ago
Rimon
33
asked 2 days ago
Rimon
33
asked 2 days ago
Rimon
33
33
Are you doing some assignment or homework regarding to the group presentation given?
– Alan Wang
2 days ago
pressntation but I don't know how to find all the subgroups as needed
– Rimon
2 days ago
2
This question was asked very recently. In fact, unless I'm mistaken this is the third time I see this question within a week. You do know that you should not repost the same question over and over again?
– Jyrki Lahtonen
2 days ago
2
You posted this. You are supposed to EDIT that first version if you have something to add to it. Reposting is simply rude, because it creates orphaned threads. It is also pointless, because anyone into group theory questions will see the version from yesterday. Why do you want to attract negative attention?
– Jyrki Lahtonen
2 days ago
ok I will edit the previous one
– Rimon
2 days ago
 |Â
show 6 more comments
Are you doing some assignment or homework regarding to the group presentation given?
– Alan Wang
2 days ago
pressntation but I don't know how to find all the subgroups as needed
– Rimon
2 days ago
2
This question was asked very recently. In fact, unless I'm mistaken this is the third time I see this question within a week. You do know that you should not repost the same question over and over again?
– Jyrki Lahtonen
2 days ago
2
You posted this. You are supposed to EDIT that first version if you have something to add to it. Reposting is simply rude, because it creates orphaned threads. It is also pointless, because anyone into group theory questions will see the version from yesterday. Why do you want to attract negative attention?
– Jyrki Lahtonen
2 days ago
ok I will edit the previous one
– Rimon
2 days ago
Are you doing some assignment or homework regarding to the group presentation given?
– Alan Wang
2 days ago
Are you doing some assignment or homework regarding to the group presentation given?
– Alan Wang
2 days ago
pressntation but I don't know how to find all the subgroups as needed
– Rimon
2 days ago
pressntation but I don't know how to find all the subgroups as needed
– Rimon
2 days ago
2
2
This question was asked very recently. In fact, unless I'm mistaken this is the third time I see this question within a week. You do know that you should not repost the same question over and over again?
– Jyrki Lahtonen
2 days ago
This question was asked very recently. In fact, unless I'm mistaken this is the third time I see this question within a week. You do know that you should not repost the same question over and over again?
– Jyrki Lahtonen
2 days ago
2
2
You posted this. You are supposed to EDIT that first version if you have something to add to it. Reposting is simply rude, because it creates orphaned threads. It is also pointless, because anyone into group theory questions will see the version from yesterday. Why do you want to attract negative attention?
– Jyrki Lahtonen
2 days ago
You posted this. You are supposed to EDIT that first version if you have something to add to it. Reposting is simply rude, because it creates orphaned threads. It is also pointless, because anyone into group theory questions will see the version from yesterday. Why do you want to attract negative attention?
– Jyrki Lahtonen
2 days ago
ok I will edit the previous one
– Rimon
2 days ago
ok I will edit the previous one
– Rimon
2 days ago
 |Â
show 6 more comments
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Are you doing some assignment or homework regarding to the group presentation given?
– Alan Wang
2 days ago
pressntation but I don't know how to find all the subgroups as needed
– Rimon
2 days ago
2
This question was asked very recently. In fact, unless I'm mistaken this is the third time I see this question within a week. You do know that you should not repost the same question over and over again?
– Jyrki Lahtonen
2 days ago
2
You posted this. You are supposed to EDIT that first version if you have something to add to it. Reposting is simply rude, because it creates orphaned threads. It is also pointless, because anyone into group theory questions will see the version from yesterday. Why do you want to attract negative attention?
– Jyrki Lahtonen
2 days ago
ok I will edit the previous one
– Rimon
2 days ago