Where do the enormous simple groups come from?
$begingroup$
I mean, these simple groups of big order such as
808017424794512875886459904961710757005754368000000000
I think it's order is something similar to a factorial for all those 0s... but I'd like to know how were these built...
abstract-algebra group-theory finite-groups simple-groups
edited Mar 8 '15 at 19:51
Matt Samuel
39.3k63870
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,440721
$endgroup$
add a comment |
$begingroup$
I mean, these simple groups of big order such as
808017424794512875886459904961710757005754368000000000
I think it's order is something similar to a factorial for all those 0s... but I'd like to know how were these built...
abstract-algebra group-theory finite-groups simple-groups
edited Mar 8 '15 at 19:51
Matt Samuel
39.3k63870
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,440721
$endgroup$
2
$begingroup$
I assume you're referring to the sporadic groups. There are finite simple groups of arbitrarily large order.
$endgroup$
– Matt Samuel
Mar 8 '15 at 19:50
1
$begingroup$
Related: What is the simplest way to fathom the Monster Group?, Why is the Monster group the largest sporadic finite simple group?.
$endgroup$
– MJD
Mar 8 '15 at 19:51
$begingroup$
I think this q/a on MathOverflow would be helpful: mathoverflow.net/questions/38161/… .
$endgroup$
– zibadawa timmy
Mar 8 '15 at 19:57
1
$begingroup$
recommend maa.org/publications/maa-reviews/…
$endgroup$
– Will Jagy
Mar 8 '15 at 20:40
add a comment |
$begingroup$
I mean, these simple groups of big order such as
808017424794512875886459904961710757005754368000000000
I think it's order is something similar to a factorial for all those 0s... but I'd like to know how were these built...
abstract-algebra group-theory finite-groups simple-groups
edited Mar 8 '15 at 19:51
Matt Samuel
39.3k63870
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,440721
$endgroup$
I mean, these simple groups of big order such as
808017424794512875886459904961710757005754368000000000
I think it's order is something similar to a factorial for all those 0s... but I'd like to know how were these built...
abstract-algebra group-theory finite-groups simple-groups
abstract-algebra group-theory finite-groups simple-groups
edited Mar 8 '15 at 19:51
Matt Samuel
39.3k63870
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,440721
edited Mar 8 '15 at 19:51
Matt Samuel
39.3k63870
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,440721
edited Mar 8 '15 at 19:51
Matt Samuel
39.3k63870
edited Mar 8 '15 at 19:51
Matt Samuel
39.3k63870
edited Mar 8 '15 at 19:51
Matt Samuel
39.3k63870
39.3k63870
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,440721
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,440721
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,440721
1,440721
2
$begingroup$
I assume you're referring to the sporadic groups. There are finite simple groups of arbitrarily large order.
$endgroup$
– Matt Samuel
Mar 8 '15 at 19:50
1
$begingroup$
Related: What is the simplest way to fathom the Monster Group?, Why is the Monster group the largest sporadic finite simple group?.
$endgroup$
– MJD
Mar 8 '15 at 19:51
$begingroup$
I think this q/a on MathOverflow would be helpful: mathoverflow.net/questions/38161/… .
$endgroup$
– zibadawa timmy
Mar 8 '15 at 19:57
1
$begingroup$
recommend maa.org/publications/maa-reviews/…
$endgroup$
– Will Jagy
Mar 8 '15 at 20:40
add a comment |
2
$begingroup$
I assume you're referring to the sporadic groups. There are finite simple groups of arbitrarily large order.
$endgroup$
– Matt Samuel
Mar 8 '15 at 19:50
1
$begingroup$
Related: What is the simplest way to fathom the Monster Group?, Why is the Monster group the largest sporadic finite simple group?.
$endgroup$
– MJD
Mar 8 '15 at 19:51
$begingroup$
I think this q/a on MathOverflow would be helpful: mathoverflow.net/questions/38161/… .
$endgroup$
– zibadawa timmy
Mar 8 '15 at 19:57
1
$begingroup$
recommend maa.org/publications/maa-reviews/…
$endgroup$
– Will Jagy
Mar 8 '15 at 20:40
2
2
$begingroup$
I assume you're referring to the sporadic groups. There are finite simple groups of arbitrarily large order.
$endgroup$
– Matt Samuel
Mar 8 '15 at 19:50
$begingroup$
I assume you're referring to the sporadic groups. There are finite simple groups of arbitrarily large order.
$endgroup$
– Matt Samuel
Mar 8 '15 at 19:50
1
1
$begingroup$
Related: What is the simplest way to fathom the Monster Group?, Why is the Monster group the largest sporadic finite simple group?.
$endgroup$
– MJD
Mar 8 '15 at 19:51
$begingroup$
Related: What is the simplest way to fathom the Monster Group?, Why is the Monster group the largest sporadic finite simple group?.
$endgroup$
– MJD
Mar 8 '15 at 19:51
$begingroup$
I think this q/a on MathOverflow would be helpful: mathoverflow.net/questions/38161/… .
$endgroup$
– zibadawa timmy
Mar 8 '15 at 19:57
$begingroup$
I think this q/a on MathOverflow would be helpful: mathoverflow.net/questions/38161/… .
$endgroup$
– zibadawa timmy
Mar 8 '15 at 19:57
1
1
$begingroup$
recommend maa.org/publications/maa-reviews/…
$endgroup$
– Will Jagy
Mar 8 '15 at 20:40
$begingroup$
recommend maa.org/publications/maa-reviews/…
$endgroup$
– Will Jagy
Mar 8 '15 at 20:40
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
(long for a comment..)
If you refer to the monster group, then I would suggest you this http://youtu.be/jsSeoGpiWsw - if John Conway says that he doesn't know why the monster group exists, I doubt that anyone can give a reasonable answer. In general, Matt Samuel's comment above is right, it is enough to consider a finite simple group of classical Lie type (i.e. certain matrices of size $n$ over a certain field) when $n$ goes to infinity the order of the group goes to infinity.
If you want a reason for why the study of finite simple groups is become important (from a theoretical point of view) then you should have a look at the Jordan-Hölder theorem, which states that every finite group is built up by a finite number of finite simple groups. Hence the finite simple groups are the elements of the 'periodic table of group theory'.
edited Jan 18 at 16:57
Martin Sleziak
45.1k10123277
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
$endgroup$
add a comment |
Your Answer
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%2f1181238%2fwhere-do-the-enormous-simple-groups-come-from%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
(long for a comment..)
If you refer to the monster group, then I would suggest you this http://youtu.be/jsSeoGpiWsw - if John Conway says that he doesn't know why the monster group exists, I doubt that anyone can give a reasonable answer. In general, Matt Samuel's comment above is right, it is enough to consider a finite simple group of classical Lie type (i.e. certain matrices of size $n$ over a certain field) when $n$ goes to infinity the order of the group goes to infinity.
If you want a reason for why the study of finite simple groups is become important (from a theoretical point of view) then you should have a look at the Jordan-Hölder theorem, which states that every finite group is built up by a finite number of finite simple groups. Hence the finite simple groups are the elements of the 'periodic table of group theory'.
edited Jan 18 at 16:57
Martin Sleziak
45.1k10123277
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
$endgroup$
add a comment |
$begingroup$
(long for a comment..)
If you refer to the monster group, then I would suggest you this http://youtu.be/jsSeoGpiWsw - if John Conway says that he doesn't know why the monster group exists, I doubt that anyone can give a reasonable answer. In general, Matt Samuel's comment above is right, it is enough to consider a finite simple group of classical Lie type (i.e. certain matrices of size $n$ over a certain field) when $n$ goes to infinity the order of the group goes to infinity.
If you want a reason for why the study of finite simple groups is become important (from a theoretical point of view) then you should have a look at the Jordan-Hölder theorem, which states that every finite group is built up by a finite number of finite simple groups. Hence the finite simple groups are the elements of the 'periodic table of group theory'.
edited Jan 18 at 16:57
Martin Sleziak
45.1k10123277
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
$endgroup$
add a comment |
$begingroup$
(long for a comment..)
If you refer to the monster group, then I would suggest you this http://youtu.be/jsSeoGpiWsw - if John Conway says that he doesn't know why the monster group exists, I doubt that anyone can give a reasonable answer. In general, Matt Samuel's comment above is right, it is enough to consider a finite simple group of classical Lie type (i.e. certain matrices of size $n$ over a certain field) when $n$ goes to infinity the order of the group goes to infinity.
If you want a reason for why the study of finite simple groups is become important (from a theoretical point of view) then you should have a look at the Jordan-Hölder theorem, which states that every finite group is built up by a finite number of finite simple groups. Hence the finite simple groups are the elements of the 'periodic table of group theory'.
edited Jan 18 at 16:57
Martin Sleziak
45.1k10123277
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
$endgroup$
(long for a comment..)
If you refer to the monster group, then I would suggest you this http://youtu.be/jsSeoGpiWsw - if John Conway says that he doesn't know why the monster group exists, I doubt that anyone can give a reasonable answer. In general, Matt Samuel's comment above is right, it is enough to consider a finite simple group of classical Lie type (i.e. certain matrices of size $n$ over a certain field) when $n$ goes to infinity the order of the group goes to infinity.
If you want a reason for why the study of finite simple groups is become important (from a theoretical point of view) then you should have a look at the Jordan-Hölder theorem, which states that every finite group is built up by a finite number of finite simple groups. Hence the finite simple groups are the elements of the 'periodic table of group theory'.
edited Jan 18 at 16:57
Martin Sleziak
45.1k10123277
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
edited Jan 18 at 16:57
Martin Sleziak
45.1k10123277
edited Jan 18 at 16:57
Martin Sleziak
45.1k10123277
edited Jan 18 at 16:57
Martin Sleziak
45.1k10123277
45.1k10123277
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
616312
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%2f1181238%2fwhere-do-the-enormous-simple-groups-come-from%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 assume you're referring to the sporadic groups. There are finite simple groups of arbitrarily large order.
$endgroup$
– Matt Samuel
Mar 8 '15 at 19:50
1
$begingroup$
Related: What is the simplest way to fathom the Monster Group?, Why is the Monster group the largest sporadic finite simple group?.
$endgroup$
– MJD
Mar 8 '15 at 19:51
$begingroup$
I think this q/a on MathOverflow would be helpful: mathoverflow.net/questions/38161/… .
$endgroup$
– zibadawa timmy
Mar 8 '15 at 19:57
1
$begingroup$
recommend maa.org/publications/maa-reviews/…
$endgroup$
– Will Jagy
Mar 8 '15 at 20:40