Quiz:Element structure of symmetric group:S3

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

Element orders and conjugacy class structure
Review the conjugacy class structure:

{What is the number of non-identity elements of the symmetric group of degree three? - 2 - 3 - 4 + 5 - 6
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 * The order of the symmetric group is 6. There is one identity element, so the number of non-identity elements is 5.

{How many elements are there of order exactly three in the symmetric group of degree three? + 2 - 3 - 4 - 5 - 6
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{Which of the following is a correct description of the conjugacy class structure of the symmetric group of degree three? - 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
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Multiplication, conjugacy and generating sets
Review the multiplication table in cycle decomposition notation: Review the multiplication table in one-line notation:

{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? - 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 product can have order either 1 or 3
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 * 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 $$(1,2,3)$$ or $$(1,3,2)$$, both of which have order three.

{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? + 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 product can have order either 1 or 3
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 * The only two elements of order three are the 3-cycles $$(1,2,3)$$ and $$(1,3,2)$$, and their product is the identity element because they are inverses of each other.

{Which of the following is false in the symmetric group of degree three? - Any two elements of the same order are conjugate - Every element is conjugate to its inverse - Any two elements generating the same cyclic subgroup are conjugate + Any two elements that together generate the whole group are conjugate - None of the above, i.e., they are all true
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 * 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.
 * 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 groups.
 * 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 groups. See symmetric groups are rational.
 * In fact, the elements $$(1,2)$$ and $$(1,2,3)$$ generate the whole group, and are not conjugate to each other.

Conjugation and commutator operations
Review the conjugation operation: Review the commutator operation: {Suppose $$g$$ and $$h$$ are distinct elements of order two in the symmetric group of order three. What can we say about $$ghg^{-1}$$ (this is a conjugate of $$h$$ by $$g$$)? - It equals $$g$$ - It equals $$h$$ + It equals an element of order two that is neither $$g$$ nor $$h$$ - It is an element of order three - It is the identity element
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{Suppose $$g$$ and $$h$$ are distinct elements of order two in the symmetric group of order three. What can we say about the commutator $$ghg^{-1}h^{-1}$$? - It equals $$g$$ - It equals $$h$$ - It equals an element of order two that is neither $$g$$ nor $$h$$ + It is an element of order three - It is the identity element
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{Suppose $$g$$ and $$h$$ are distinct elements of order three in the symmetric group of order three. What can we say about $$ghg^{-1}$$ (this is a conjugate of $$h$$ by $$g$$)? - It equals $$g$$ + It equals $$h$$ - It equals an element of order two - It is the identity element
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{Suppose $$g$$ and $$h$$ are distinct elements of order three in the symmetric group of order three. What can we say about the commutator $$ghg^{-1}h^{-1}$$? - It equals $$g$$ - It equals $$h$$ - It is an element of order two + It is the identity element
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{Suppose $$g$$ is an element of order two and $$h$$ is an element of order three in the symmetric group of order three. What are the orders of the elements $$ghg^{-1}$$ and $$hgh^{-1}$$ respectively? - 1 and 1 - 2 and 3 + 3 and 2 - 2 and 2 - 3 and 3
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