Sizes of Conjugacy Classes of a Group of Known Order
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Suppose G is a group of order 48 (centre consisting identity only). Show it has a conjugacy class of order 3.
I know that the size of the conjugacy classes are limited to divisors of 48: 1,2,3,4,6,8,12,16,24, and 48. These classes also partition G so their sizes must sum to the order of G (so I cannot use 48 and have to include 1). I am not sure how to continue from here.
abstract-algebra group-theory permutations equivalence-relations
asked Nov 18 at 22:53
J. Dawson
143
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1
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Suppose G is a group of order 48 (centre consisting identity only). Show it has a conjugacy class of order 3.
I know that the size of the conjugacy classes are limited to divisors of 48: 1,2,3,4,6,8,12,16,24, and 48. These classes also partition G so their sizes must sum to the order of G (so I cannot use 48 and have to include 1). I am not sure how to continue from here.
abstract-algebra group-theory permutations equivalence-relations
asked Nov 18 at 22:53
J. Dawson
143
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose G is a group of order 48 (centre consisting identity only). Show it has a conjugacy class of order 3.
I know that the size of the conjugacy classes are limited to divisors of 48: 1,2,3,4,6,8,12,16,24, and 48. These classes also partition G so their sizes must sum to the order of G (so I cannot use 48 and have to include 1). I am not sure how to continue from here.
abstract-algebra group-theory permutations equivalence-relations
asked Nov 18 at 22:53
J. Dawson
143
Suppose G is a group of order 48 (centre consisting identity only). Show it has a conjugacy class of order 3.
I know that the size of the conjugacy classes are limited to divisors of 48: 1,2,3,4,6,8,12,16,24, and 48. These classes also partition G so their sizes must sum to the order of G (so I cannot use 48 and have to include 1). I am not sure how to continue from here.
abstract-algebra group-theory permutations equivalence-relations
abstract-algebra group-theory permutations equivalence-relations
asked Nov 18 at 22:53
J. Dawson
143
asked Nov 18 at 22:53
J. Dawson
143
edited Nov 19 at 0:27
asked Nov 18 at 22:53
J. Dawson
143
asked Nov 18 at 22:53
J. Dawson
143
asked Nov 18 at 22:53
J. Dawson
143
143
add a comment |
add a comment |
1 Answer
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We know that $G$ partitions into conjugacy classes and the only possible orders of these classes are those that you mention. There is only one conjugacy class of order $1$ since the center is trivial. Now the sum of the occurring orders is $48$, which is even, so there is at least one conjugacy class of odd order bigger than one...
answered Nov 18 at 23:16
Jef L
2,745617
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
We know that $G$ partitions into conjugacy classes and the only possible orders of these classes are those that you mention. There is only one conjugacy class of order $1$ since the center is trivial. Now the sum of the occurring orders is $48$, which is even, so there is at least one conjugacy class of odd order bigger than one...
answered Nov 18 at 23:16
Jef L
2,745617
add a comment |
up vote
4
down vote
We know that $G$ partitions into conjugacy classes and the only possible orders of these classes are those that you mention. There is only one conjugacy class of order $1$ since the center is trivial. Now the sum of the occurring orders is $48$, which is even, so there is at least one conjugacy class of odd order bigger than one...
answered Nov 18 at 23:16
Jef L
2,745617
add a comment |
up vote
4
down vote
up vote
4
down vote
We know that $G$ partitions into conjugacy classes and the only possible orders of these classes are those that you mention. There is only one conjugacy class of order $1$ since the center is trivial. Now the sum of the occurring orders is $48$, which is even, so there is at least one conjugacy class of odd order bigger than one...
answered Nov 18 at 23:16
Jef L
2,745617
We know that $G$ partitions into conjugacy classes and the only possible orders of these classes are those that you mention. There is only one conjugacy class of order $1$ since the center is trivial. Now the sum of the occurring orders is $48$, which is even, so there is at least one conjugacy class of odd order bigger than one...
answered Nov 18 at 23:16
Jef L
2,745617
answered Nov 18 at 23:16
Jef L
2,745617
answered Nov 18 at 23:16
Jef L
2,745617
answered Nov 18 at 23:16
Jef L
2,745617
2,745617
add a comment |
add a comment |
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