Is there a common notation for characteristic subgroups?
$begingroup$
There is a common notation in group theory for normal subgroups. Also there is a notation for subnormal subgroups. But quite often we deal with characteristic subgroups. So, my question
Is there a common notation for characteristic subgroups? And if there is no common notation, is this a good idea to denote a characteristic subgroup $H$ of group $G$ in the following manner $H operatorname{char} G$?
abstract-algebra group-theory notation
asked Dec 21 '18 at 9:10
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
$endgroup$
|
show 1 more comment
$begingroup$
There is a common notation in group theory for normal subgroups. Also there is a notation for subnormal subgroups. But quite often we deal with characteristic subgroups. So, my question
Is there a common notation for characteristic subgroups? And if there is no common notation, is this a good idea to denote a characteristic subgroup $H$ of group $G$ in the following manner $H operatorname{char} G$?
abstract-algebra group-theory notation
asked Dec 21 '18 at 9:10
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
$endgroup$
2
$begingroup$
I encountered the notation that you proposed in "An introduction to the Theory of Groups" of Joseph J. Rotman. So feel free to use it, I would say.
$endgroup$
– drhab
Dec 21 '18 at 9:39
$begingroup$
I also think that $H operatorname{char} G$ is most widely understood. Just as an addition: My algebra professor many years ago used a symbol derived from the "normal subgroup" triangle, but with an extra vertical line on the right (something like $triangleleftsmall|$). I kind of liked that symbol, but I have never seen it in use since (and LaTeX and MathJax don't support it either), so I believe my professer just made it up.
$endgroup$
– jpvee
Dec 21 '18 at 14:22
$begingroup$
I was taught when writing maths to always say something (within reason) to back up your notation. So for example, don't write "Let $Hoperatorname{char}G$", but instead write something like "Let $Hoperatorname{char}G$ be a characteristic subgroup of $G$". However, personally I would not use this notation the notation unless it is really necessary and just write "Let $Hleq G$ be a characteristic subgroup of $G$".
$endgroup$
– user1729
Dec 21 '18 at 14:24
1
$begingroup$
I'd suggest $Hunderset{mathrm{char}}{lhd}G$. It makes it clear that $H$ is a normal subgroup of $G$, and then clear that it's meant to be characteristic.
$endgroup$
– YCor
Dec 21 '18 at 14:44
$begingroup$
@user1729, sometimes it is convinient to write sometithing as $H operatorname{char} N vartriangleleft G$ so $H vartriangleleft G$
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 16:15
|
show 1 more comment
$begingroup$
There is a common notation in group theory for normal subgroups. Also there is a notation for subnormal subgroups. But quite often we deal with characteristic subgroups. So, my question
Is there a common notation for characteristic subgroups? And if there is no common notation, is this a good idea to denote a characteristic subgroup $H$ of group $G$ in the following manner $H operatorname{char} G$?
abstract-algebra group-theory notation
asked Dec 21 '18 at 9:10
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
$endgroup$
There is a common notation in group theory for normal subgroups. Also there is a notation for subnormal subgroups. But quite often we deal with characteristic subgroups. So, my question
Is there a common notation for characteristic subgroups? And if there is no common notation, is this a good idea to denote a characteristic subgroup $H$ of group $G$ in the following manner $H operatorname{char} G$?
abstract-algebra group-theory notation
abstract-algebra group-theory notation
asked Dec 21 '18 at 9:10
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 21 '18 at 9:10
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 21 '18 at 9:10
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 21 '18 at 9:10
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 21 '18 at 9:10
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
623414
2
$begingroup$
I encountered the notation that you proposed in "An introduction to the Theory of Groups" of Joseph J. Rotman. So feel free to use it, I would say.
$endgroup$
– drhab
Dec 21 '18 at 9:39
$begingroup$
I also think that $H operatorname{char} G$ is most widely understood. Just as an addition: My algebra professor many years ago used a symbol derived from the "normal subgroup" triangle, but with an extra vertical line on the right (something like $triangleleftsmall|$). I kind of liked that symbol, but I have never seen it in use since (and LaTeX and MathJax don't support it either), so I believe my professer just made it up.
$endgroup$
– jpvee
Dec 21 '18 at 14:22
$begingroup$
I was taught when writing maths to always say something (within reason) to back up your notation. So for example, don't write "Let $Hoperatorname{char}G$", but instead write something like "Let $Hoperatorname{char}G$ be a characteristic subgroup of $G$". However, personally I would not use this notation the notation unless it is really necessary and just write "Let $Hleq G$ be a characteristic subgroup of $G$".
$endgroup$
– user1729
Dec 21 '18 at 14:24
1
$begingroup$
I'd suggest $Hunderset{mathrm{char}}{lhd}G$. It makes it clear that $H$ is a normal subgroup of $G$, and then clear that it's meant to be characteristic.
$endgroup$
– YCor
Dec 21 '18 at 14:44
$begingroup$
@user1729, sometimes it is convinient to write sometithing as $H operatorname{char} N vartriangleleft G$ so $H vartriangleleft G$
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 16:15
|
show 1 more comment
2
$begingroup$
I encountered the notation that you proposed in "An introduction to the Theory of Groups" of Joseph J. Rotman. So feel free to use it, I would say.
$endgroup$
– drhab
Dec 21 '18 at 9:39
$begingroup$
I also think that $H operatorname{char} G$ is most widely understood. Just as an addition: My algebra professor many years ago used a symbol derived from the "normal subgroup" triangle, but with an extra vertical line on the right (something like $triangleleftsmall|$). I kind of liked that symbol, but I have never seen it in use since (and LaTeX and MathJax don't support it either), so I believe my professer just made it up.
$endgroup$
– jpvee
Dec 21 '18 at 14:22
$begingroup$
I was taught when writing maths to always say something (within reason) to back up your notation. So for example, don't write "Let $Hoperatorname{char}G$", but instead write something like "Let $Hoperatorname{char}G$ be a characteristic subgroup of $G$". However, personally I would not use this notation the notation unless it is really necessary and just write "Let $Hleq G$ be a characteristic subgroup of $G$".
$endgroup$
– user1729
Dec 21 '18 at 14:24
1
$begingroup$
I'd suggest $Hunderset{mathrm{char}}{lhd}G$. It makes it clear that $H$ is a normal subgroup of $G$, and then clear that it's meant to be characteristic.
$endgroup$
– YCor
Dec 21 '18 at 14:44
$begingroup$
@user1729, sometimes it is convinient to write sometithing as $H operatorname{char} N vartriangleleft G$ so $H vartriangleleft G$
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 16:15
2
2
$begingroup$
I encountered the notation that you proposed in "An introduction to the Theory of Groups" of Joseph J. Rotman. So feel free to use it, I would say.
$endgroup$
– drhab
Dec 21 '18 at 9:39
$begingroup$
I encountered the notation that you proposed in "An introduction to the Theory of Groups" of Joseph J. Rotman. So feel free to use it, I would say.
$endgroup$
– drhab
Dec 21 '18 at 9:39
$begingroup$
I also think that $H operatorname{char} G$ is most widely understood. Just as an addition: My algebra professor many years ago used a symbol derived from the "normal subgroup" triangle, but with an extra vertical line on the right (something like $triangleleftsmall|$). I kind of liked that symbol, but I have never seen it in use since (and LaTeX and MathJax don't support it either), so I believe my professer just made it up.
$endgroup$
– jpvee
Dec 21 '18 at 14:22
$begingroup$
I also think that $H operatorname{char} G$ is most widely understood. Just as an addition: My algebra professor many years ago used a symbol derived from the "normal subgroup" triangle, but with an extra vertical line on the right (something like $triangleleftsmall|$). I kind of liked that symbol, but I have never seen it in use since (and LaTeX and MathJax don't support it either), so I believe my professer just made it up.
$endgroup$
– jpvee
Dec 21 '18 at 14:22
$begingroup$
I was taught when writing maths to always say something (within reason) to back up your notation. So for example, don't write "Let $Hoperatorname{char}G$", but instead write something like "Let $Hoperatorname{char}G$ be a characteristic subgroup of $G$". However, personally I would not use this notation the notation unless it is really necessary and just write "Let $Hleq G$ be a characteristic subgroup of $G$".
$endgroup$
– user1729
Dec 21 '18 at 14:24
$begingroup$
I was taught when writing maths to always say something (within reason) to back up your notation. So for example, don't write "Let $Hoperatorname{char}G$", but instead write something like "Let $Hoperatorname{char}G$ be a characteristic subgroup of $G$". However, personally I would not use this notation the notation unless it is really necessary and just write "Let $Hleq G$ be a characteristic subgroup of $G$".
$endgroup$
– user1729
Dec 21 '18 at 14:24
1
1
$begingroup$
I'd suggest $Hunderset{mathrm{char}}{lhd}G$. It makes it clear that $H$ is a normal subgroup of $G$, and then clear that it's meant to be characteristic.
$endgroup$
– YCor
Dec 21 '18 at 14:44
$begingroup$
I'd suggest $Hunderset{mathrm{char}}{lhd}G$. It makes it clear that $H$ is a normal subgroup of $G$, and then clear that it's meant to be characteristic.
$endgroup$
– YCor
Dec 21 '18 at 14:44
$begingroup$
@user1729, sometimes it is convinient to write sometithing as $H operatorname{char} N vartriangleleft G$ so $H vartriangleleft G$
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 16:15
$begingroup$
@user1729, sometimes it is convinient to write sometithing as $H operatorname{char} N vartriangleleft G$ so $H vartriangleleft G$
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 16:15
|
show 1 more comment
2 Answers
2
active
oldest
votes
$begingroup$
I have seen $Ntriangleleft G$ for normal subgroups and $Nblacktriangleleft G$ for characteristic subgroups. This also makes clear that a characteristic subgroup is normal.
answered Dec 21 '18 at 16:29
Dietrich BurdeDietrich Burde
79.3k647103
$endgroup$
$begingroup$
There is a potential issue here that if someone is just flicking through a paper than they might just assume that this notation meant "normal".
$endgroup$
– user1729
Dec 21 '18 at 20:34
$begingroup$
@user1729 Yes, you are right, there are also several arguments against it, like that it is not used by many people, but I find it different enough from normal subgroup, because it is filled with black.
$endgroup$
– Dietrich Burde
Dec 21 '18 at 21:14
$begingroup$
I will use this notation in my notes)
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 17:07
add a comment |
$begingroup$
There is no common notation for this subgroup, in the sense, not all algebra text has a common notation. However, some authors use H char G to denote H is the characteristic subgroup of G. For example, Dummit and Foote use this notation. Also from the comment, Rotman use this notation. So from now on, use this notation. There's no problem!
answered Dec 21 '18 at 12:07
Chinnapparaj RChinnapparaj R
5,5072928
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f3048320%2fis-there-a-common-notation-for-characteristic-subgroups%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I have seen $Ntriangleleft G$ for normal subgroups and $Nblacktriangleleft G$ for characteristic subgroups. This also makes clear that a characteristic subgroup is normal.
answered Dec 21 '18 at 16:29
Dietrich BurdeDietrich Burde
79.3k647103
$endgroup$
$begingroup$
There is a potential issue here that if someone is just flicking through a paper than they might just assume that this notation meant "normal".
$endgroup$
– user1729
Dec 21 '18 at 20:34
$begingroup$
@user1729 Yes, you are right, there are also several arguments against it, like that it is not used by many people, but I find it different enough from normal subgroup, because it is filled with black.
$endgroup$
– Dietrich Burde
Dec 21 '18 at 21:14
$begingroup$
I will use this notation in my notes)
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 17:07
add a comment |
$begingroup$
I have seen $Ntriangleleft G$ for normal subgroups and $Nblacktriangleleft G$ for characteristic subgroups. This also makes clear that a characteristic subgroup is normal.
answered Dec 21 '18 at 16:29
Dietrich BurdeDietrich Burde
79.3k647103
$endgroup$
$begingroup$
There is a potential issue here that if someone is just flicking through a paper than they might just assume that this notation meant "normal".
$endgroup$
– user1729
Dec 21 '18 at 20:34
$begingroup$
@user1729 Yes, you are right, there are also several arguments against it, like that it is not used by many people, but I find it different enough from normal subgroup, because it is filled with black.
$endgroup$
– Dietrich Burde
Dec 21 '18 at 21:14
$begingroup$
I will use this notation in my notes)
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 17:07
add a comment |
$begingroup$
I have seen $Ntriangleleft G$ for normal subgroups and $Nblacktriangleleft G$ for characteristic subgroups. This also makes clear that a characteristic subgroup is normal.
answered Dec 21 '18 at 16:29
Dietrich BurdeDietrich Burde
79.3k647103
$endgroup$
I have seen $Ntriangleleft G$ for normal subgroups and $Nblacktriangleleft G$ for characteristic subgroups. This also makes clear that a characteristic subgroup is normal.
answered Dec 21 '18 at 16:29
Dietrich BurdeDietrich Burde
79.3k647103
answered Dec 21 '18 at 16:29
Dietrich BurdeDietrich Burde
79.3k647103
answered Dec 21 '18 at 16:29
Dietrich BurdeDietrich Burde
79.3k647103
answered Dec 21 '18 at 16:29
Dietrich BurdeDietrich Burde
79.3k647103
79.3k647103
$begingroup$
There is a potential issue here that if someone is just flicking through a paper than they might just assume that this notation meant "normal".
$endgroup$
– user1729
Dec 21 '18 at 20:34
$begingroup$
@user1729 Yes, you are right, there are also several arguments against it, like that it is not used by many people, but I find it different enough from normal subgroup, because it is filled with black.
$endgroup$
– Dietrich Burde
Dec 21 '18 at 21:14
$begingroup$
I will use this notation in my notes)
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 17:07
add a comment |
$begingroup$
There is a potential issue here that if someone is just flicking through a paper than they might just assume that this notation meant "normal".
$endgroup$
– user1729
Dec 21 '18 at 20:34
$begingroup$
@user1729 Yes, you are right, there are also several arguments against it, like that it is not used by many people, but I find it different enough from normal subgroup, because it is filled with black.
$endgroup$
– Dietrich Burde
Dec 21 '18 at 21:14
$begingroup$
I will use this notation in my notes)
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 17:07
$begingroup$
There is a potential issue here that if someone is just flicking through a paper than they might just assume that this notation meant "normal".
$endgroup$
– user1729
Dec 21 '18 at 20:34
$begingroup$
There is a potential issue here that if someone is just flicking through a paper than they might just assume that this notation meant "normal".
$endgroup$
– user1729
Dec 21 '18 at 20:34
$begingroup$
@user1729 Yes, you are right, there are also several arguments against it, like that it is not used by many people, but I find it different enough from normal subgroup, because it is filled with black.
$endgroup$
– Dietrich Burde
Dec 21 '18 at 21:14
$begingroup$
@user1729 Yes, you are right, there are also several arguments against it, like that it is not used by many people, but I find it different enough from normal subgroup, because it is filled with black.
$endgroup$
– Dietrich Burde
Dec 21 '18 at 21:14
$begingroup$
I will use this notation in my notes)
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 17:07
$begingroup$
I will use this notation in my notes)
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 17:07
add a comment |
$begingroup$
There is no common notation for this subgroup, in the sense, not all algebra text has a common notation. However, some authors use H char G to denote H is the characteristic subgroup of G. For example, Dummit and Foote use this notation. Also from the comment, Rotman use this notation. So from now on, use this notation. There's no problem!
answered Dec 21 '18 at 12:07
Chinnapparaj RChinnapparaj R
5,5072928
$endgroup$
add a comment |
$begingroup$
There is no common notation for this subgroup, in the sense, not all algebra text has a common notation. However, some authors use H char G to denote H is the characteristic subgroup of G. For example, Dummit and Foote use this notation. Also from the comment, Rotman use this notation. So from now on, use this notation. There's no problem!
answered Dec 21 '18 at 12:07
Chinnapparaj RChinnapparaj R
5,5072928
$endgroup$
add a comment |
$begingroup$
There is no common notation for this subgroup, in the sense, not all algebra text has a common notation. However, some authors use H char G to denote H is the characteristic subgroup of G. For example, Dummit and Foote use this notation. Also from the comment, Rotman use this notation. So from now on, use this notation. There's no problem!
answered Dec 21 '18 at 12:07
Chinnapparaj RChinnapparaj R
5,5072928
$endgroup$
There is no common notation for this subgroup, in the sense, not all algebra text has a common notation. However, some authors use H char G to denote H is the characteristic subgroup of G. For example, Dummit and Foote use this notation. Also from the comment, Rotman use this notation. So from now on, use this notation. There's no problem!
answered Dec 21 '18 at 12:07
Chinnapparaj RChinnapparaj R
5,5072928
answered Dec 21 '18 at 12:07
Chinnapparaj RChinnapparaj R
5,5072928
answered Dec 21 '18 at 12:07
Chinnapparaj RChinnapparaj R
5,5072928
answered Dec 21 '18 at 12:07
Chinnapparaj RChinnapparaj R
5,5072928
5,5072928
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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%2f3048320%2fis-there-a-common-notation-for-characteristic-subgroups%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
2
$begingroup$
I encountered the notation that you proposed in "An introduction to the Theory of Groups" of Joseph J. Rotman. So feel free to use it, I would say.
$endgroup$
– drhab
Dec 21 '18 at 9:39
$begingroup$
I also think that $H operatorname{char} G$ is most widely understood. Just as an addition: My algebra professor many years ago used a symbol derived from the "normal subgroup" triangle, but with an extra vertical line on the right (something like $triangleleftsmall|$). I kind of liked that symbol, but I have never seen it in use since (and LaTeX and MathJax don't support it either), so I believe my professer just made it up.
$endgroup$
– jpvee
Dec 21 '18 at 14:22
$begingroup$
I was taught when writing maths to always say something (within reason) to back up your notation. So for example, don't write "Let $Hoperatorname{char}G$", but instead write something like "Let $Hoperatorname{char}G$ be a characteristic subgroup of $G$". However, personally I would not use this notation the notation unless it is really necessary and just write "Let $Hleq G$ be a characteristic subgroup of $G$".
$endgroup$
– user1729
Dec 21 '18 at 14:24
1
$begingroup$
I'd suggest $Hunderset{mathrm{char}}{lhd}G$. It makes it clear that $H$ is a normal subgroup of $G$, and then clear that it's meant to be characteristic.
$endgroup$
– YCor
Dec 21 '18 at 14:44
$begingroup$
@user1729, sometimes it is convinient to write sometithing as $H operatorname{char} N vartriangleleft G$ so $H vartriangleleft G$
$endgroup$
– Mikhail Goltvanitsa
Dec 21 '18 at 16:15