Let n \ge 7. The conjugacy class of fact about::transpositions in ... This fails for n = 6, where the conjugacy class of triple transpositions ... ...
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5, such that P_1 and P_2 have the same conjugacy class size statistics but ... Element structure of groups of order 32#Conjugacy class sizes ... ...
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#redirect conjugacy class size statistics need not determine nilpotency class for groups of prime-fifth order ... ...
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repetitions, i.e., if a particular conjugacy class size occurs for multiple ... Then, the conjugacy class sizes for the external direct product are ... ...
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conjugacy classes but have different conjugacy class size statistics. ... * Nilpotency class and order need not determine conjugacy class size ... ...
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class c of the group determines its conjugacy class size statistics. ... * Nilpotency class and order need not determine conjugacy class size ... ...
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nilpotency class but have different conjugacy class size statistics. ... * Conjugacy class size statistics need not determine nilpotency class ... ...
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A conjugacy class in a group can be defined in any of the following ways: ... Suppose G is a group. A conjugacy class in G is a nonempty subset ... ...
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Let (G,\Phi) be an APS of groups. The conjugacy class APS of G is ... by this equivalence relation gives the conjugacy class APS. ... ...
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Let G be a defining ingredient::FC-group (i.e., every conjugacy class ... are as follows: the coefficient of conjugacy class k in the product of ... ...
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Let H be a subgroup of G. A coset-conjugacy class of H is the set ... coset of H lie in the same coset-conjugacy class. In fact, two left cosets ... ...
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#redirect Conjugacy class size set of a finite group ... ...
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Let G be a group. A conjugacy class of G is called a self-inverse conjugacy class if each element in the group is conjugate to its inverse, that ... ...
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The term real conjugacy class is somewhat ambiguous and could refer to either of two things:
* A conjugacy class of real elements, i.e., a conjugacy ... ...
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The term rational conjugacy class is ambiguous and could refer to either of the following things:
* A conjugacy class of rational elements, i ... ...
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#redirect conjugacy class size statistics of a finite group ... ...
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#redirect Determination of conjugacy class structure of symmetric group:S3 ... ...
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For G a group, the maximum conjugacy class size is the largest size ... * The maximum conjugacy class size of any abelian group G is 1, since ... ...
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large for the finite group, a conjugacy class-representation bijection ... A conjugacy class-representation bijection is said to be characteristic ... ...
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This article describes the duality between conjugacy classes and representations, as well as puts important results of character theory in the ... ...
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Let n \ge 7. The conjugacy class of fact about::transpositions in ... This fails for n = 6, where the conjugacy class of triple transpositions ... ...
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Let (G,\Phi) be an APS of groups. The conjugacy class APS of G is ... by this equivalence relation gives the conjugacy class APS. ... ...
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The term real conjugacy class is somewhat ambiguous and could refer to either of two things:
* A conjugacy class of real elements, i.e., a conjugacy ... ...
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* The corresponding conjugacy class APS is cancellative, in the sense ... * Call the conjugacy class of g \in G_m indecomposable if no element ... ...
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Let G be a finite group. The conjugacy class size statistics of G ... The conjugacy class size statistics carry more information than the ... ...
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of a group G and g \in H. Then, the conjugacy class of g with respect to ... H-conjugacy classes in the G-conjugacy class of g \leftrightarrow ... ...
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groups G_1 and G_2 such that the conjugacy class size statistics of G_1 ... representations need not determine conjugacy class size statistics ... ...
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A finite group G is said to be conjugacy class size set-determined if whenever H is a group such that:
* G and H have the same order (i.e. cardinality) ... ...
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===Element orders and conjugacy class structure=== Review the conjugacy class structure:
conjugacy class structure ... ...
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nilpotency class but have different conjugacy class size statistics. ... * Conjugacy class size statistics need not determine nilpotency class ... ...
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==Statement==
The following are equivalent for a p-Sylow subgroup P of a finite group G:
# P is a normal subgroup of G and possesses a normal ... ...
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For G a group, the maximum conjugacy class size is the largest size ... * The maximum conjugacy class size of any abelian group G is 1, since ... ...
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conjugacy classes but have different conjugacy class size statistics. ... * Nilpotency class and order need not determine conjugacy class size ... ...
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The size of the centralizer of a conjugacy class with i_j cycles of ... Hence, the size of the conjugacy class is:
\frac{n!}{\prod_j (j)^{i_j} i_j!} ... ...
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G_1 and G_2 such that the multisets of conjugacy class size statistics are ... of groups of order 32#Grouping by conjugacy class sizes. Note that the ... ...
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c_1) and \rho(c_2) are not in the same conjugacy class in GL(V). ... ===Definition in terms of the conjugacy class-representation duality=== ... ...
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large for the finite group, a conjugacy class-representation bijection ... A conjugacy class-representation bijection is said to be characteristic ... ...
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Let G be a group. A conjugacy class of G is called a self-inverse conjugacy class if each element in the group is conjugate to its inverse, that ... ...
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general linear group, are in the same conjugacy class of the general linear group. ... Let L(c,\rho) denote the conjugacy class of \rho(g) in the general ... ...
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repetitions, i.e., if a particular conjugacy class size occurs for multiple ... Then, the conjugacy class sizes for the external direct product are ... ...
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