Difference between revisions of "Quiz:Symmetric group:S3"

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(Elements)
(Elements)
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==Elements==
 
==Elements==
  
See [[element structure of symmetric group:S3]] for full details.
+
[[:Quiz:Element structure of symmetric group:S3]]
 
 
Review the multiplication table in cycle decomposition notation: <toggledisplay>{{#lst:element structure of symmetric group:S3|multiplication table}}</toggledisplay>
 
<br>
 
Review the multiplication table in one-line notation: <toggledisplay>{{#lst:element structure of symmetric group:S3|multiplication table in one-line notation}}</toggledisplay>
 
<br>
 
Review the conjugacy class structure: <toggledisplay>{{#lst:element structure of symmetric group:S3|conjugacy class structure}}</toggledisplay>
 
 
 
<quiz display=simple>
 
{What is the number of non-identity elements of the symmetric group of degree three?
 
|type="()"}
 
- 2
 
- 3
 
- 4
 
+ 5
 
|| The order of the symmetric group is 6. There is one identity element, so the number of non-identity elements is 5.
 
- 6
 
 
 
{How many elements are there of order exactly three in the symmetric group of degree three?
 
|type="()"}
 
+ 2
 
- 3
 
- 4
 
- 5
 
- 6
 
 
 
{What can we say about the order of the product of two ''distinct'' elements, each of order exactly two, in the symmetric group of degree three?
 
|type="()"}
 
- The product must be the identity element
 
- The product must have order two
 
- The product can have order either 1 or 2
 
+ The product must have order three
 
|| The two distinct elements are distinct 2-transpositions, and are both odd, hence their product is an even permutation. However, they are distinct, and each of them is its own inverse, so the product of distinct elements cannot be the identity element. Hence, it is one of the 3-cycles <math>(1,2,3)</math> or <math>(1,3,2)</math>, both of which have order three.
 
- The product can have order either 1 or 3
 
 
 
{What can we say about the order of the product of two ''distinct'' elements, each of order exactly three, in the symmetric group of degree three?
 
|type="()"}
 
+ The product must be the identity element
 
|| The only two elements of order three are the 3-cycles <math>(1,2,3)</math> and <matH>(1,3,2)</math>, and their product is the identity element because they are inverses of each other.
 
- The product must have order two
 
- The product can have order either 1 or 2
 
- The product must have order three
 
- The product can have order either 1 or 3
 
 
 
{Which of the following is ''false'' in the symmetric group of degree three?
 
|type="()"}
 
- Any two elements of the same order are conjugate
 
|| See [[group in which every element is order-conjugate]]. Note that among finite groups, the only groups with this property are [[trivial group]], [[cyclic group:Z2]], and [[symmetric group:S3]].
 
- Every element is conjugate to its inverse
 
|| Follows by the fact that [[cycle type determines conjugacy class]] and every element has the same cycle type as its inverse. All symmetric groups have this property. Groups with this property are termed [[ambivalent group]]s.
 
- Any two elements generating the same cyclic subgroup are conjugate
 
|| Follows by the fact that [[cycle type determines conjugacy class]] and any two elements generating the same cyclic subgroup have the same cycle type. All symmetric groups have this property. Groups with this property are termed [[rational group]]s. See [[symmetric groups are rational]].
 
+ Any two elements that together generate the whole group are conjugate
 
|| In fact, the elements <math>(1,2)</math> and <math>(1,2,3)</math> generate the whole group, and are not conjugate to each other.
 
- None of the above, i.e., they are all true
 
 
 
{Which of the following is a correct description of the conjugacy class structure of the symmetric group of degree three?
 
|type="()"}
 
- Conjugacy class of size 1 and order (of elements in the conjugacy class) 1, conjugacy class of size 2 and order (of elements in the conjugacy class) 2, conjugacy class of size 3 and order (of elements in the conjugacy class) 3
 
+ Conjugacy class of size 1 and order (of elements in the conjugacy class) 1, conjugacy class of size 2 and order (of elements in the conjugacy class) 3, conjugacy class of size 3 and order (of elements in the conjugacy class) 2
 
- Conjugacy class of size 1 and order (of elements in the conjugacy class) 2, conjugacy class of size 2 and order (of elements in the conjugacy class) 1, conjugacy class of size 3 and order (of elements in the conjugacy class) 3
 
- Conjugacy class of size 1 and order (of elements in the conjugacy class) 2, conjugacy class of size 2 and order (of elements in the conjugacy class) 3, conjugacy class of size 3 and order (of elements in the conjugacy class) 1
 
- Conjugacy class of size 1 and order (of elements in the conjugacy class) 3, conjugacy class of size 2 and order (of elements in the conjugacy class) 1, conjugacy class of size 3 and order (of elements in the conjugacy class) 2
 
- Conjugacy class of size 1 and order (of elements in the conjugacy class) 3, conjugacy class of size 2 and order (of elements in the conjugacy class) 2, conjugacy class of size 3 and order (of elements in the conjugacy class) 1
 
 
 
</quiz>
 
  
 
==Subgroups==
 
==Subgroups==

Revision as of 15:06, 8 October 2011

See symmetric group:S3. We take the symmetric group on the set \{ 1,2,3 \} of size three.

Elements

Quiz:Element structure of symmetric group:S3

Subgroups

Summary table on the structure of subgroups: [SHOW MORE]

1 What are the possible orders of subgroups of the symmetric group of degree three?

1 and 6 only
1, 2, and 6 only
1, 3, and 6 only
1, 2, 3, and 6 only
1, 4, and 6 only
1, 2, 3, 4, 5, and 6 only

2 What are the possible orders of normal subgroups of the symmetric group of degree three?

1 and 6 only
1, 2, and 6 only
1, 3, and 6 only
1, 2, 3, and 6 only
1, 4, and 6 only
1, 2, 3, 4, 5, and 6 only