Let $a$ and $b$ be elements of a group $G$, and $H$ and $K$ be subgroups of $G$. If $aH=bK$, prove that...
up vote
2
down vote
favorite
Let $a$ and $b$ be elements of a group $G$ and $H$ and $K$ be subgroups of $G$. If $aH=bK$, prove that $H=K$.
My attempt: we know that $aH=bH$ if $ain bH$ or $a^{-1}bin H$. I think that these properties will be used in the solution, but I cannot understand how they will be used since we not know if $a$ and $b$ are part of any subgroup.
I thought of proving somehow that $aH=H$ and $bK=K$ so we could show $H=K$, but again I am stuck at the fact that I do not know if $a$ belongs to $H$ or not.
What should be my next steps?
group-theory
asked Nov 13 at 13:38
Gaurang Tandon
3,48522147
add a comment |
up vote
2
down vote
favorite
Let $a$ and $b$ be elements of a group $G$ and $H$ and $K$ be subgroups of $G$. If $aH=bK$, prove that $H=K$.
My attempt: we know that $aH=bH$ if $ain bH$ or $a^{-1}bin H$. I think that these properties will be used in the solution, but I cannot understand how they will be used since we not know if $a$ and $b$ are part of any subgroup.
I thought of proving somehow that $aH=H$ and $bK=K$ so we could show $H=K$, but again I am stuck at the fact that I do not know if $a$ belongs to $H$ or not.
What should be my next steps?
group-theory
asked Nov 13 at 13:38
Gaurang Tandon
3,48522147
Hint: prove that $a^{-1}bin Hcap K$.
– user10354138
Nov 13 at 13:50
@user10354138 Sorry I couldn't get it to work :(
– Gaurang Tandon
Nov 13 at 14:07
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $a$ and $b$ be elements of a group $G$ and $H$ and $K$ be subgroups of $G$. If $aH=bK$, prove that $H=K$.
My attempt: we know that $aH=bH$ if $ain bH$ or $a^{-1}bin H$. I think that these properties will be used in the solution, but I cannot understand how they will be used since we not know if $a$ and $b$ are part of any subgroup.
I thought of proving somehow that $aH=H$ and $bK=K$ so we could show $H=K$, but again I am stuck at the fact that I do not know if $a$ belongs to $H$ or not.
What should be my next steps?
group-theory
asked Nov 13 at 13:38
Gaurang Tandon
3,48522147
Let $a$ and $b$ be elements of a group $G$ and $H$ and $K$ be subgroups of $G$. If $aH=bK$, prove that $H=K$.
My attempt: we know that $aH=bH$ if $ain bH$ or $a^{-1}bin H$. I think that these properties will be used in the solution, but I cannot understand how they will be used since we not know if $a$ and $b$ are part of any subgroup.
I thought of proving somehow that $aH=H$ and $bK=K$ so we could show $H=K$, but again I am stuck at the fact that I do not know if $a$ belongs to $H$ or not.
What should be my next steps?
group-theory
group-theory
asked Nov 13 at 13:38
Gaurang Tandon
3,48522147
asked Nov 13 at 13:38
Gaurang Tandon
3,48522147
asked Nov 13 at 13:38
Gaurang Tandon
3,48522147
asked Nov 13 at 13:38
Gaurang Tandon
3,48522147
asked Nov 13 at 13:38
Gaurang Tandon
3,48522147
3,48522147
Hint: prove that $a^{-1}bin Hcap K$.
– user10354138
Nov 13 at 13:50
@user10354138 Sorry I couldn't get it to work :(
– Gaurang Tandon
Nov 13 at 14:07
add a comment |
Hint: prove that $a^{-1}bin Hcap K$.
– user10354138
Nov 13 at 13:50
@user10354138 Sorry I couldn't get it to work :(
– Gaurang Tandon
Nov 13 at 14:07
Hint: prove that $a^{-1}bin Hcap K$.
– user10354138
Nov 13 at 13:50
Hint: prove that $a^{-1}bin Hcap K$.
– user10354138
Nov 13 at 13:50
@user10354138 Sorry I couldn't get it to work :(
– Gaurang Tandon
Nov 13 at 14:07
@user10354138 Sorry I couldn't get it to work :(
– Gaurang Tandon
Nov 13 at 14:07
add a comment |
2 Answers
2
active
oldest
votes
up vote
4
down vote
accepted
Simply see that if $ae=a in aH$, so $ain bK implies aK=bK=aH$
Now, multiply by $a^{-1}$ both sides and you have $H=K$
answered Nov 13 at 14:33
Lord KK
78628
add a comment |
up vote
1
down vote
$textbf{Hint:}$
$e$ is in both $H$ and $K$ so $ae=bk_1$. Similarly for other cases. Then try to show the same thing common between both in terms of $a$ and $b$. Then from equality, you get whatever you wanted to prove.
edited 2 days ago
amWhy
191k27223437
answered Nov 13 at 14:02
Shubham
1,2931518
From here I get $a=bk_1$ and $b=ah_1$, which yields that $ab^{-1}=k_1$ and $a^{-1}b=h_1$. What do i do now?
– Gaurang Tandon
Nov 13 at 14:06
What is relations with 2 element you have found?
– Shubham
Nov 13 at 14:15
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
Simply see that if $ae=a in aH$, so $ain bK implies aK=bK=aH$
Now, multiply by $a^{-1}$ both sides and you have $H=K$
answered Nov 13 at 14:33
Lord KK
78628
add a comment |
up vote
4
down vote
accepted
Simply see that if $ae=a in aH$, so $ain bK implies aK=bK=aH$
Now, multiply by $a^{-1}$ both sides and you have $H=K$
answered Nov 13 at 14:33
Lord KK
78628
add a comment |
up vote
4
down vote
accepted
up vote
4
down vote
accepted
Simply see that if $ae=a in aH$, so $ain bK implies aK=bK=aH$
Now, multiply by $a^{-1}$ both sides and you have $H=K$
answered Nov 13 at 14:33
Lord KK
78628
Simply see that if $ae=a in aH$, so $ain bK implies aK=bK=aH$
Now, multiply by $a^{-1}$ both sides and you have $H=K$
answered Nov 13 at 14:33
Lord KK
78628
answered Nov 13 at 14:33
Lord KK
78628
answered Nov 13 at 14:33
Lord KK
78628
answered Nov 13 at 14:33
Lord KK
78628
78628
add a comment |
add a comment |
up vote
1
down vote
$textbf{Hint:}$
$e$ is in both $H$ and $K$ so $ae=bk_1$. Similarly for other cases. Then try to show the same thing common between both in terms of $a$ and $b$. Then from equality, you get whatever you wanted to prove.
edited 2 days ago
amWhy
191k27223437
answered Nov 13 at 14:02
Shubham
1,2931518
From here I get $a=bk_1$ and $b=ah_1$, which yields that $ab^{-1}=k_1$ and $a^{-1}b=h_1$. What do i do now?
– Gaurang Tandon
Nov 13 at 14:06
What is relations with 2 element you have found?
– Shubham
Nov 13 at 14:15
add a comment |
up vote
1
down vote
$textbf{Hint:}$
$e$ is in both $H$ and $K$ so $ae=bk_1$. Similarly for other cases. Then try to show the same thing common between both in terms of $a$ and $b$. Then from equality, you get whatever you wanted to prove.
edited 2 days ago
amWhy
191k27223437
answered Nov 13 at 14:02
Shubham
1,2931518
From here I get $a=bk_1$ and $b=ah_1$, which yields that $ab^{-1}=k_1$ and $a^{-1}b=h_1$. What do i do now?
– Gaurang Tandon
Nov 13 at 14:06
What is relations with 2 element you have found?
– Shubham
Nov 13 at 14:15
add a comment |
up vote
1
down vote
up vote
1
down vote
$textbf{Hint:}$
$e$ is in both $H$ and $K$ so $ae=bk_1$. Similarly for other cases. Then try to show the same thing common between both in terms of $a$ and $b$. Then from equality, you get whatever you wanted to prove.
edited 2 days ago
amWhy
191k27223437
answered Nov 13 at 14:02
Shubham
1,2931518
$textbf{Hint:}$
$e$ is in both $H$ and $K$ so $ae=bk_1$. Similarly for other cases. Then try to show the same thing common between both in terms of $a$ and $b$. Then from equality, you get whatever you wanted to prove.
edited 2 days ago
amWhy
191k27223437
answered Nov 13 at 14:02
Shubham
1,2931518
edited 2 days ago
amWhy
191k27223437
edited 2 days ago
amWhy
191k27223437
edited 2 days ago
amWhy
191k27223437
191k27223437
answered Nov 13 at 14:02
Shubham
1,2931518
answered Nov 13 at 14:02
Shubham
1,2931518
answered Nov 13 at 14:02
Shubham
1,2931518
1,2931518
From here I get $a=bk_1$ and $b=ah_1$, which yields that $ab^{-1}=k_1$ and $a^{-1}b=h_1$. What do i do now?
– Gaurang Tandon
Nov 13 at 14:06
What is relations with 2 element you have found?
– Shubham
Nov 13 at 14:15
add a comment |
From here I get $a=bk_1$ and $b=ah_1$, which yields that $ab^{-1}=k_1$ and $a^{-1}b=h_1$. What do i do now?
– Gaurang Tandon
Nov 13 at 14:06
What is relations with 2 element you have found?
– Shubham
Nov 13 at 14:15
From here I get $a=bk_1$ and $b=ah_1$, which yields that $ab^{-1}=k_1$ and $a^{-1}b=h_1$. What do i do now?
– Gaurang Tandon
Nov 13 at 14:06
From here I get $a=bk_1$ and $b=ah_1$, which yields that $ab^{-1}=k_1$ and $a^{-1}b=h_1$. What do i do now?
– Gaurang Tandon
Nov 13 at 14:06
What is relations with 2 element you have found?
– Shubham
Nov 13 at 14:15
What is relations with 2 element you have found?
– Shubham
Nov 13 at 14:15
add a comment |
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2996739%2flet-a-and-b-be-elements-of-a-group-g-and-h-and-k-be-subgroups-of-g%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Hint: prove that $a^{-1}bin Hcap K$.
– user10354138
Nov 13 at 13:50
@user10354138 Sorry I couldn't get it to work :(
– Gaurang Tandon
Nov 13 at 14:07