Simple groups of the same order
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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 at 15:57
Jim Humphreys
41.4k491188
asked Dec 7 at 14:53
Mare
3,57121231
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up vote
7
down vote
favorite
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 at 15:57
Jim Humphreys
41.4k491188
asked Dec 7 at 14:53
Mare
3,57121231
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?
– David Richter
Dec 7 at 17:17
add a comment |
up vote
7
down vote
favorite
up vote
7
down vote
favorite
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 at 15:57
Jim Humphreys
41.4k491188
asked Dec 7 at 14:53
Mare
3,57121231
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 at 15:57
Jim Humphreys
41.4k491188
asked Dec 7 at 14:53
Mare
3,57121231
edited Dec 7 at 15:57
Jim Humphreys
41.4k491188
asked Dec 7 at 14:53
Mare
3,57121231
edited Dec 7 at 15:57
Jim Humphreys
41.4k491188
edited Dec 7 at 15:57
Jim Humphreys
41.4k491188
edited Dec 7 at 15:57
Jim Humphreys
41.4k491188
41.4k491188
asked Dec 7 at 14:53
Mare
3,57121231
asked Dec 7 at 14:53
Mare
3,57121231
asked Dec 7 at 14:53
Mare
3,57121231
3,57121231
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?
– David Richter
Dec 7 at 17:17
add a comment |
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?
– David Richter
Dec 7 at 17:17
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?
– David Richter
Dec 7 at 17:17
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?
– David Richter
Dec 7 at 17:17
add a comment |
1 Answer
1
active
oldest
votes
up vote
13
down vote
accepted
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 at 15:54
Jim Humphreys
41.4k491188
answered Dec 7 at 15:09
Derek Holt
26.4k461108
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
13
down vote
accepted
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 at 15:54
Jim Humphreys
41.4k491188
answered Dec 7 at 15:09
Derek Holt
26.4k461108
add a comment |
up vote
13
down vote
accepted
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 at 15:54
Jim Humphreys
41.4k491188
answered Dec 7 at 15:09
Derek Holt
26.4k461108
add a comment |
up vote
13
down vote
accepted
up vote
13
down vote
accepted
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 at 15:54
Jim Humphreys
41.4k491188
answered Dec 7 at 15:09
Derek Holt
26.4k461108
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 at 15:54
Jim Humphreys
41.4k491188
answered Dec 7 at 15:09
Derek Holt
26.4k461108
edited Dec 7 at 15:54
Jim Humphreys
41.4k491188
edited Dec 7 at 15:54
Jim Humphreys
41.4k491188
edited Dec 7 at 15:54
Jim Humphreys
41.4k491188
41.4k491188
answered Dec 7 at 15:09
Derek Holt
26.4k461108
answered Dec 7 at 15:09
Derek Holt
26.4k461108
answered Dec 7 at 15:09
Derek Holt
26.4k461108
26.4k461108
add a comment |
add a comment |
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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?
– David Richter
Dec 7 at 17:17