Simple groups of the same order
$begingroup$
I heard that there are no 3 nonisomorphic simple groups of the same order.
Question: Is there an elementary proof of this?
In case this is not the case, here a modified question:
Question: Is there an elementary proof that there are not $m$ nonisomorphic simple groups of the same order with $m geq 4$ as small as possible?
gr.group-theory finite-groups
edited Dec 7 '18 at 15:57
Jim Humphreys
41.6k493190
asked Dec 7 '18 at 14:53
MareMare
3,57821234
$endgroup$
add a comment |
$begingroup$
I heard that there are no 3 nonisomorphic simple groups of the same order.
Question: Is there an elementary proof of this?
In case this is not the case, here a modified question:
Question: Is there an elementary proof that there are not $m$ nonisomorphic simple groups of the same order with $m geq 4$ as small as possible?
gr.group-theory finite-groups
edited Dec 7 '18 at 15:57
Jim Humphreys
41.6k493190
asked Dec 7 '18 at 14:53
MareMare
3,57821234
$endgroup$
$begingroup$
A related question: How spread out are the (non-abelian) simple groups in terms of their orders? What are some "near misses"? For example, 168-60 is already quite small. Are there smaller differences, perhaps scaling to account for size?
$endgroup$
– David Richter
Dec 7 '18 at 17:17
add a comment |
$begingroup$
I heard that there are no 3 nonisomorphic simple groups of the same order.
Question: Is there an elementary proof of this?
In case this is not the case, here a modified question:
Question: Is there an elementary proof that there are not $m$ nonisomorphic simple groups of the same order with $m geq 4$ as small as possible?
gr.group-theory finite-groups
edited Dec 7 '18 at 15:57
Jim Humphreys
41.6k493190
asked Dec 7 '18 at 14:53
MareMare
3,57821234
$endgroup$
I heard that there are no 3 nonisomorphic simple groups of the same order.
Question: Is there an elementary proof of this?
In case this is not the case, here a modified question:
Question: Is there an elementary proof that there are not $m$ nonisomorphic simple groups of the same order with $m geq 4$ as small as possible?
gr.group-theory finite-groups
gr.group-theory finite-groups
edited Dec 7 '18 at 15:57
Jim Humphreys
41.6k493190
asked Dec 7 '18 at 14:53
MareMare
3,57821234
edited Dec 7 '18 at 15:57
Jim Humphreys
41.6k493190
asked Dec 7 '18 at 14:53
MareMare
3,57821234
edited Dec 7 '18 at 15:57
Jim Humphreys
41.6k493190
edited Dec 7 '18 at 15:57
Jim Humphreys
41.6k493190
edited Dec 7 '18 at 15:57
Jim Humphreys
41.6k493190
41.6k493190
asked Dec 7 '18 at 14:53
MareMare
3,57821234
asked Dec 7 '18 at 14:53
MareMare
3,57821234
asked Dec 7 '18 at 14:53
MareMare
3,57821234
3,57821234
$begingroup$
A related question: How spread out are the (non-abelian) simple groups in terms of their orders? What are some "near misses"? For example, 168-60 is already quite small. Are there smaller differences, perhaps scaling to account for size?
$endgroup$
– David Richter
Dec 7 '18 at 17:17
add a comment |
$begingroup$
A related question: How spread out are the (non-abelian) simple groups in terms of their orders? What are some "near misses"? For example, 168-60 is already quite small. Are there smaller differences, perhaps scaling to account for size?
$endgroup$
– David Richter
Dec 7 '18 at 17:17
$begingroup$
A related question: How spread out are the (non-abelian) simple groups in terms of their orders? What are some "near misses"? For example, 168-60 is already quite small. Are there smaller differences, perhaps scaling to account for size?
$endgroup$
– David Richter
Dec 7 '18 at 17:17
$begingroup$
A related question: How spread out are the (non-abelian) simple groups in terms of their orders? What are some "near misses"? For example, 168-60 is already quite small. Are there smaller differences, perhaps scaling to account for size?
$endgroup$
– David Richter
Dec 7 '18 at 17:17
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, there are no known proofs of any results of this type that do not rely on the complete classification of finite simple groups
In particular, the result of Pyber (1993) giving an upper bound on the number of isomorphism classes of finite groups of order $n$ (see Jack Schmidt's answer to this question for details) could only be incorrect if there were vast numbers of isomorphism classes of simple groups of the same order, but it still relies on the classification.
edited Dec 7 '18 at 15:54
Jim Humphreys
41.6k493190
answered Dec 7 '18 at 15:09
Derek HoltDerek Holt
26.8k462110
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
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oldest
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$begingroup$
No, there are no known proofs of any results of this type that do not rely on the complete classification of finite simple groups
In particular, the result of Pyber (1993) giving an upper bound on the number of isomorphism classes of finite groups of order $n$ (see Jack Schmidt's answer to this question for details) could only be incorrect if there were vast numbers of isomorphism classes of simple groups of the same order, but it still relies on the classification.
edited Dec 7 '18 at 15:54
Jim Humphreys
41.6k493190
answered Dec 7 '18 at 15:09
Derek HoltDerek Holt
26.8k462110
$endgroup$
add a comment |
$begingroup$
No, there are no known proofs of any results of this type that do not rely on the complete classification of finite simple groups
In particular, the result of Pyber (1993) giving an upper bound on the number of isomorphism classes of finite groups of order $n$ (see Jack Schmidt's answer to this question for details) could only be incorrect if there were vast numbers of isomorphism classes of simple groups of the same order, but it still relies on the classification.
edited Dec 7 '18 at 15:54
Jim Humphreys
41.6k493190
answered Dec 7 '18 at 15:09
Derek HoltDerek Holt
26.8k462110
$endgroup$
add a comment |
$begingroup$
No, there are no known proofs of any results of this type that do not rely on the complete classification of finite simple groups
In particular, the result of Pyber (1993) giving an upper bound on the number of isomorphism classes of finite groups of order $n$ (see Jack Schmidt's answer to this question for details) could only be incorrect if there were vast numbers of isomorphism classes of simple groups of the same order, but it still relies on the classification.
edited Dec 7 '18 at 15:54
Jim Humphreys
41.6k493190
answered Dec 7 '18 at 15:09
Derek HoltDerek Holt
26.8k462110
$endgroup$
No, there are no known proofs of any results of this type that do not rely on the complete classification of finite simple groups
In particular, the result of Pyber (1993) giving an upper bound on the number of isomorphism classes of finite groups of order $n$ (see Jack Schmidt's answer to this question for details) could only be incorrect if there were vast numbers of isomorphism classes of simple groups of the same order, but it still relies on the classification.
edited Dec 7 '18 at 15:54
Jim Humphreys
41.6k493190
answered Dec 7 '18 at 15:09
Derek HoltDerek Holt
26.8k462110
edited Dec 7 '18 at 15:54
Jim Humphreys
41.6k493190
edited Dec 7 '18 at 15:54
Jim Humphreys
41.6k493190
edited Dec 7 '18 at 15:54
Jim Humphreys
41.6k493190
41.6k493190
answered Dec 7 '18 at 15:09
Derek HoltDerek Holt
26.8k462110
answered Dec 7 '18 at 15:09
Derek HoltDerek Holt
26.8k462110
answered Dec 7 '18 at 15:09
Derek HoltDerek Holt
26.8k462110
26.8k462110
add a comment |
add a comment |
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$begingroup$
A related question: How spread out are the (non-abelian) simple groups in terms of their orders? What are some "near misses"? For example, 168-60 is already quite small. Are there smaller differences, perhaps scaling to account for size?
$endgroup$
– David Richter
Dec 7 '18 at 17:17