Mixing
Find two finite groups $G_1$ and $G_2$ such as:
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Find two finite groups $G_1$ and $G_2$ such as:
$1)$ $|G_1|=|G_2|$
$2)$ for all prime $p$ every $p$-sylow subgroup of $G_1$ isomorpic to $p$-sylow subgroup of $G_2$.
but $G_1notcong G_2$.
hello I tried to find an example but I couldn't find one please help me.
I look on all the groups from order 6 (I choose this order randomly) but I don't know how to prove that every p-sylow subgroup of $G_1$ isomorpic to every p-sylow subgroup of $G_2$ for all primes p.
finite-groups sylow-theory
asked Aug 3 at 11:44
Rimon
33
add a comment |Â
up vote
1
down vote
favorite
Find two finite groups $G_1$ and $G_2$ such as:
$1)$ $|G_1|=|G_2|$
$2)$ for all prime $p$ every $p$-sylow subgroup of $G_1$ isomorpic to $p$-sylow subgroup of $G_2$.
but $G_1notcong G_2$.
hello I tried to find an example but I couldn't find one please help me.
I look on all the groups from order 6 (I choose this order randomly) but I don't know how to prove that every p-sylow subgroup of $G_1$ isomorpic to every p-sylow subgroup of $G_2$ for all primes p.
finite-groups sylow-theory
asked Aug 3 at 11:44
Rimon
33
Maybe try to add what you have already tried.
– zzuussee
Aug 3 at 11:46
1
Have you seen semi-direct product? If so $C_21$ and $C_3ltimes C_7$ would do.
– daruma
Aug 3 at 11:49
The first version of your question was not very detailed, but your edit is already better. You can take a look at this page to find some advice about what you should put in your question : math.meta.stackexchange.com/questions/9959/…
– Arnaud D.
Aug 3 at 11:55
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Find two finite groups $G_1$ and $G_2$ such as:
$1)$ $|G_1|=|G_2|$
$2)$ for all prime $p$ every $p$-sylow subgroup of $G_1$ isomorpic to $p$-sylow subgroup of $G_2$.
but $G_1notcong G_2$.
hello I tried to find an example but I couldn't find one please help me.
I look on all the groups from order 6 (I choose this order randomly) but I don't know how to prove that every p-sylow subgroup of $G_1$ isomorpic to every p-sylow subgroup of $G_2$ for all primes p.
finite-groups sylow-theory
asked Aug 3 at 11:44
Rimon
33
Find two finite groups $G_1$ and $G_2$ such as:
$1)$ $|G_1|=|G_2|$
$2)$ for all prime $p$ every $p$-sylow subgroup of $G_1$ isomorpic to $p$-sylow subgroup of $G_2$.
but $G_1notcong G_2$.
hello I tried to find an example but I couldn't find one please help me.
I look on all the groups from order 6 (I choose this order randomly) but I don't know how to prove that every p-sylow subgroup of $G_1$ isomorpic to every p-sylow subgroup of $G_2$ for all primes p.
finite-groups sylow-theory
asked Aug 3 at 11:44
Rimon
33
edited Aug 3 at 11:52
asked Aug 3 at 11:44
Rimon
33
asked Aug 3 at 11:44
Rimon
33
asked Aug 3 at 11:44
Rimon
33
33
Maybe try to add what you have already tried.
– zzuussee
Aug 3 at 11:46
1
Have you seen semi-direct product? If so $C_21$ and $C_3ltimes C_7$ would do.
– daruma
Aug 3 at 11:49
The first version of your question was not very detailed, but your edit is already better. You can take a look at this page to find some advice about what you should put in your question : math.meta.stackexchange.com/questions/9959/…
– Arnaud D.
Aug 3 at 11:55
add a comment |Â
Maybe try to add what you have already tried.
– zzuussee
Aug 3 at 11:46
1
Have you seen semi-direct product? If so $C_21$ and $C_3ltimes C_7$ would do.
– daruma
Aug 3 at 11:49
The first version of your question was not very detailed, but your edit is already better. You can take a look at this page to find some advice about what you should put in your question : math.meta.stackexchange.com/questions/9959/…
– Arnaud D.
Aug 3 at 11:55
Maybe try to add what you have already tried.
– zzuussee
Aug 3 at 11:46
Maybe try to add what you have already tried.
– zzuussee
Aug 3 at 11:46
1
1
Have you seen semi-direct product? If so $C_21$ and $C_3ltimes C_7$ would do.
– daruma
Aug 3 at 11:49
Have you seen semi-direct product? If so $C_21$ and $C_3ltimes C_7$ would do.
– daruma
Aug 3 at 11:49
The first version of your question was not very detailed, but your edit is already better. You can take a look at this page to find some advice about what you should put in your question : math.meta.stackexchange.com/questions/9959/…
– Arnaud D.
Aug 3 at 11:55
The first version of your question was not very detailed, but your edit is already better. You can take a look at this page to find some advice about what you should put in your question : math.meta.stackexchange.com/questions/9959/…
– Arnaud D.
Aug 3 at 11:55
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
If $p, q$ are distinct primes then every two groups of order $pq$ will have isomorphic Sylow subgroups (being cyclic subgroups of order $p$ and $q$). So all you have to do is to find two nonisomorphic subgroups of order $pq$, e.g. $mathbbZ_6$ and $S_3$.
answered Aug 3 at 11:50
freakish
8,4371524
thank you :) ,I am sorry but how do I prove that every p-sylow subgroups ane isomorpic?
– Rimon
Aug 3 at 11:58
1
@Rimon Every two cyclic groups of the same order are isomorphic. And every group of order $pq$ has both: subgroups of order $p$ and of order $q$ (Cauchy's theorem). These subgroups are necessary Sylow. That's everything you need.
– freakish
Aug 3 at 12:01
thank you :) I get it now
– Rimon
Aug 3 at 12:30
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
If $p, q$ are distinct primes then every two groups of order $pq$ will have isomorphic Sylow subgroups (being cyclic subgroups of order $p$ and $q$). So all you have to do is to find two nonisomorphic subgroups of order $pq$, e.g. $mathbbZ_6$ and $S_3$.
answered Aug 3 at 11:50
freakish
8,4371524
thank you :) ,I am sorry but how do I prove that every p-sylow subgroups ane isomorpic?
– Rimon
Aug 3 at 11:58
1
@Rimon Every two cyclic groups of the same order are isomorphic. And every group of order $pq$ has both: subgroups of order $p$ and of order $q$ (Cauchy's theorem). These subgroups are necessary Sylow. That's everything you need.
– freakish
Aug 3 at 12:01
thank you :) I get it now
– Rimon
Aug 3 at 12:30
add a comment |Â
up vote
4
down vote
If $p, q$ are distinct primes then every two groups of order $pq$ will have isomorphic Sylow subgroups (being cyclic subgroups of order $p$ and $q$). So all you have to do is to find two nonisomorphic subgroups of order $pq$, e.g. $mathbbZ_6$ and $S_3$.
answered Aug 3 at 11:50
freakish
8,4371524
thank you :) ,I am sorry but how do I prove that every p-sylow subgroups ane isomorpic?
– Rimon
Aug 3 at 11:58
1
@Rimon Every two cyclic groups of the same order are isomorphic. And every group of order $pq$ has both: subgroups of order $p$ and of order $q$ (Cauchy's theorem). These subgroups are necessary Sylow. That's everything you need.
– freakish
Aug 3 at 12:01
thank you :) I get it now
– Rimon
Aug 3 at 12:30
add a comment |Â
up vote
4
down vote
up vote
4
down vote
If $p, q$ are distinct primes then every two groups of order $pq$ will have isomorphic Sylow subgroups (being cyclic subgroups of order $p$ and $q$). So all you have to do is to find two nonisomorphic subgroups of order $pq$, e.g. $mathbbZ_6$ and $S_3$.
answered Aug 3 at 11:50
freakish
8,4371524
If $p, q$ are distinct primes then every two groups of order $pq$ will have isomorphic Sylow subgroups (being cyclic subgroups of order $p$ and $q$). So all you have to do is to find two nonisomorphic subgroups of order $pq$, e.g. $mathbbZ_6$ and $S_3$.
answered Aug 3 at 11:50
freakish
8,4371524
answered Aug 3 at 11:50
freakish
8,4371524
answered Aug 3 at 11:50
freakish
8,4371524
answered Aug 3 at 11:50
freakish
8,4371524
8,4371524
thank you :) ,I am sorry but how do I prove that every p-sylow subgroups ane isomorpic?
– Rimon
Aug 3 at 11:58
1
@Rimon Every two cyclic groups of the same order are isomorphic. And every group of order $pq$ has both: subgroups of order $p$ and of order $q$ (Cauchy's theorem). These subgroups are necessary Sylow. That's everything you need.
– freakish
Aug 3 at 12:01
thank you :) I get it now
– Rimon
Aug 3 at 12:30
add a comment |Â
thank you :) ,I am sorry but how do I prove that every p-sylow subgroups ane isomorpic?
– Rimon
Aug 3 at 11:58
1
@Rimon Every two cyclic groups of the same order are isomorphic. And every group of order $pq$ has both: subgroups of order $p$ and of order $q$ (Cauchy's theorem). These subgroups are necessary Sylow. That's everything you need.
– freakish
Aug 3 at 12:01
thank you :) I get it now
– Rimon
Aug 3 at 12:30
thank you :) ,I am sorry but how do I prove that every p-sylow subgroups ane isomorpic?
– Rimon
Aug 3 at 11:58
thank you :) ,I am sorry but how do I prove that every p-sylow subgroups ane isomorpic?
– Rimon
Aug 3 at 11:58
1
1
@Rimon Every two cyclic groups of the same order are isomorphic. And every group of order $pq$ has both: subgroups of order $p$ and of order $q$ (Cauchy's theorem). These subgroups are necessary Sylow. That's everything you need.
– freakish
Aug 3 at 12:01
@Rimon Every two cyclic groups of the same order are isomorphic. And every group of order $pq$ has both: subgroups of order $p$ and of order $q$ (Cauchy's theorem). These subgroups are necessary Sylow. That's everything you need.
– freakish
Aug 3 at 12:01
thank you :) I get it now
– Rimon
Aug 3 at 12:30
thank you :) I get it now
– Rimon
Aug 3 at 12:30
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2870979%2ffind-two-finite-groups-g-1-and-g-2-such-as%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Maybe try to add what you have already tried.
– zzuussee
Aug 3 at 11:46
1
Have you seen semi-direct product? If so $C_21$ and $C_3ltimes C_7$ would do.
– daruma
Aug 3 at 11:49
The first version of your question was not very detailed, but your edit is already better. You can take a look at this page to find some advice about what you should put in your question : math.meta.stackexchange.com/questions/9959/…
– Arnaud D.
Aug 3 at 11:55