Quiz:Symmetric group:S3 - Revision history

Vipul: /* Subgroups */

2011-12-29T01:36:54Z

Subgroups

← Older revision		Revision as of 01:36, 29 December 2011
Line 8:		Line 8:

	{{:Quiz:Subgroup structure of symmetric group:S3}}		{{:Quiz:Subgroup structure of symmetric group:S3}}

			==Linear representations==

			{{:Quiz:Linear representation theory of symmetric group:S3}}

Vipul: /* Subgroups */

2011-10-08T15:09:42Z

Subgroups

← Older revision		Revision as of 15:09, 8 October 2011
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	==Subgroups==		==Subgroups==

	~~Summary table on the structure of subgroups: <toggledisplay>~~{{~~#lst~~:~~subgroup~~ structure of symmetric group:S3~~\|summary}}</toggledisplay>~~		{{:Quiz:Subgroup structure of symmetric group:S3}}

	~~<quiz display=simple>~~
	~~{What are the possible orders of subgroups of the symmetric group of degree three?~~
	~~\|type="()"~~}
	~~- 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~~

	~~{What are the possible orders of normal subgroups of the symmetric group of degree three?~~
	~~\|type="()"~~}
	~~- 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~~

	~~</quiz>~~

Vipul: /* Elements */

2011-10-08T15:07:11Z

Elements

← Older revision		Revision as of 15:07, 8 October 2011
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	==Elements==		==Elements==

	[[:Quiz:Element structure of symmetric group:S3]]		{{:Quiz:Element structure of symmetric group:S3}}

	==Subgroups==		==Subgroups==

Vipul: /* Elements */

2011-10-08T15:06:52Z

Elements

@@ Line 3: / Line 3: @@
 ==Elements==
-See [[element structure of symmetric group:S3]] for full details.
+[[:Quiz:Element structure of symmetric group:S3]]
 ==Subgroups==

Vipul: /* Elements */

2011-10-08T15:03:02Z

Elements

@@ Line 46: / Line 46: @@
 - The product must have order three
 - The product can have order either 1 or 3
 </quiz>

Vipul: /* Elements */

2011-10-08T14:45:54Z

Elements

← Older revision		Revision as of 14:45, 8 October 2011
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	See [[element structure of symmetric group:S3]] for full details.		See [[element structure of symmetric group:S3]] for full details.

	Review the multiplication table: <toggledisplay>{{#lst:element structure of symmetric group:S3\|multiplication table}}</toggledisplay>		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>		Review the conjugacy class structure: <toggledisplay>{{#lst:element structure of symmetric group:S3\|conjugacy class structure}}</toggledisplay>

Vipul: Created page with "See symmetric group:S3. We take the symmetric group on the set $\{ 1,2,3 \}$ of size three. ==Elements== See element structure of symmetric group:S3 for full..."

2011-10-08T14:41:18Z

Created page with "See symmetric group:S3. We take the symmetric group on the set <math>\{ 1,2,3 \}</math> of size three. ==Elements== See element structure of symmetric group:S3 for full..."

New page

See [[symmetric group:S3]]. We take the symmetric group on the set <math>\{ 1,2,3 \}</math> of size three.

==Elements==

See [[element structure of symmetric group:S3]] for full details.

Review the multiplication table: <toggledisplay>{{#lst:element structure of symmetric group:S3|multiplication table}}</toggledisplay>

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

</quiz>

Quiz:Symmetric group:S3 - Revision history

Vipul: /* Subgroups */

Vipul: /* Subgroups */

Vipul: /* Elements */

Vipul: /* Elements */

Vipul: /* Elements */

Vipul: /* Elements */

Vipul: Created page with "See symmetric group:S3. We take the symmetric group on the set \{ 1,2,3 \} of size three. ==Elements== See element structure of symmetric group:S3 for full..."

Vipul: Created page with "See symmetric group:S3. We take the symmetric group on the set $\{ 1,2,3 \}$ of size three. ==Elements== See element structure of symmetric group:S3 for full..."