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
38.4k63768
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,368720
$endgroup$
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
38.4k63768
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,368720
$endgroup$
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
38.4k63768
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,368720
$endgroup$
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
38.4k63768
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,368720
edited Mar 8 '15 at 19:51
Matt Samuel
38.4k63768
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,368720
edited Mar 8 '15 at 19:51
Matt Samuel
38.4k63768
edited Mar 8 '15 at 19:51
Matt Samuel
38.4k63768
edited Mar 8 '15 at 19:51
Matt Samuel
38.4k63768
38.4k63768
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,368720
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,368720
asked Mar 8 '15 at 19:48
David MolanoDavid Molano
1,368720
1,368720
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
44.8k10118272
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
$endgroup$
add a comment |
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$begingroup$
(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
44.8k10118272
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
44.8k10118272
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
44.8k10118272
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
44.8k10118272
answered Mar 9 '15 at 12:54
rafforafforafforaffo
616312
edited Jan 18 at 16:57
Martin Sleziak
44.8k10118272
edited Jan 18 at 16:57
Martin Sleziak
44.8k10118272
edited Jan 18 at 16:57
Martin Sleziak
44.8k10118272
44.8k10118272
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 |
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$begingroup$
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