The cases in proving that a group of order 90 is not simple
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I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a contradiction, but why is the second case there? In other words, why do these two cases exhaust all possibilities?
abstract-algebra group-theory sylow-theory simple-groups
asked Dec 8 '18 at 22:57
WesleyWesley
518313
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add a comment |
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I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a contradiction, but why is the second case there? In other words, why do these two cases exhaust all possibilities?
abstract-algebra group-theory sylow-theory simple-groups
asked Dec 8 '18 at 22:57
WesleyWesley
518313
$endgroup$
$begingroup$
Slightly different context, but essentially a duplicate of math.stackexchange.com/questions/2536201/…
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– Eric Wofsey
Dec 8 '18 at 23:00
2
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Of course they are exhaustive: (b) is the negation of (a). If two different subgroups of order 9 do not intersect trivially, then they necessarily intersect in a 3-element subgroup, as a proper nontrivial subgroup in a 9-element group must have 3 elements by the Lagrange theorem.
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– A. Pongrácz
Dec 8 '18 at 23:04
add a comment |
$begingroup$
I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a contradiction, but why is the second case there? In other words, why do these two cases exhaust all possibilities?
abstract-algebra group-theory sylow-theory simple-groups
asked Dec 8 '18 at 22:57
WesleyWesley
518313
$endgroup$
I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a contradiction, but why is the second case there? In other words, why do these two cases exhaust all possibilities?
abstract-algebra group-theory sylow-theory simple-groups
abstract-algebra group-theory sylow-theory simple-groups
asked Dec 8 '18 at 22:57
WesleyWesley
518313
asked Dec 8 '18 at 22:57
WesleyWesley
518313
asked Dec 8 '18 at 22:57
WesleyWesley
518313
asked Dec 8 '18 at 22:57
WesleyWesley
518313
asked Dec 8 '18 at 22:57
WesleyWesley
518313
518313
$begingroup$
Slightly different context, but essentially a duplicate of math.stackexchange.com/questions/2536201/…
$endgroup$
– Eric Wofsey
Dec 8 '18 at 23:00
2
$begingroup$
Of course they are exhaustive: (b) is the negation of (a). If two different subgroups of order 9 do not intersect trivially, then they necessarily intersect in a 3-element subgroup, as a proper nontrivial subgroup in a 9-element group must have 3 elements by the Lagrange theorem.
$endgroup$
– A. Pongrácz
Dec 8 '18 at 23:04
add a comment |
$begingroup$
Slightly different context, but essentially a duplicate of math.stackexchange.com/questions/2536201/…
$endgroup$
– Eric Wofsey
Dec 8 '18 at 23:00
2
$begingroup$
Of course they are exhaustive: (b) is the negation of (a). If two different subgroups of order 9 do not intersect trivially, then they necessarily intersect in a 3-element subgroup, as a proper nontrivial subgroup in a 9-element group must have 3 elements by the Lagrange theorem.
$endgroup$
– A. Pongrácz
Dec 8 '18 at 23:04
$begingroup$
Slightly different context, but essentially a duplicate of math.stackexchange.com/questions/2536201/…
$endgroup$
– Eric Wofsey
Dec 8 '18 at 23:00
$begingroup$
Slightly different context, but essentially a duplicate of math.stackexchange.com/questions/2536201/…
$endgroup$
– Eric Wofsey
Dec 8 '18 at 23:00
2
2
$begingroup$
Of course they are exhaustive: (b) is the negation of (a). If two different subgroups of order 9 do not intersect trivially, then they necessarily intersect in a 3-element subgroup, as a proper nontrivial subgroup in a 9-element group must have 3 elements by the Lagrange theorem.
$endgroup$
– A. Pongrácz
Dec 8 '18 at 23:04
$begingroup$
Of course they are exhaustive: (b) is the negation of (a). If two different subgroups of order 9 do not intersect trivially, then they necessarily intersect in a 3-element subgroup, as a proper nontrivial subgroup in a 9-element group must have 3 elements by the Lagrange theorem.
$endgroup$
– A. Pongrácz
Dec 8 '18 at 23:04
add a comment |
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$begingroup$
Slightly different context, but essentially a duplicate of math.stackexchange.com/questions/2536201/…
$endgroup$
– Eric Wofsey
Dec 8 '18 at 23:00
2
$begingroup$
Of course they are exhaustive: (b) is the negation of (a). If two different subgroups of order 9 do not intersect trivially, then they necessarily intersect in a 3-element subgroup, as a proper nontrivial subgroup in a 9-element group must have 3 elements by the Lagrange theorem.
$endgroup$
– A. Pongrácz
Dec 8 '18 at 23:04