It is possible to have a finite group such that the fact about::field ... * Field generated by character values is splitting field implies it ... ...
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Suppose G is a group and k is a field whose characteristic does not ... # k contains a subfield K that is isomorphic to the field generated ... ...
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dividing the order of G. Suppose K is the field generated by the values of ... Then, if K is a fact about::splitting field for G (i.e., all linear ... ...
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or a prime not dividing the order of G. The field generated by character values ... * In characteristic zero, field generated by character values is contained ... ...
1 KB (169 words) - 00:55, 1 February 2013
It is possible to have a finite group G such that the field generated ... * Minimal splitting field need not be cyclotomic
* Minimal splitting ... ...
2 KB (344 words) - 00:46, 2 March 2012
Suppose G is a finite group and K is the field generated by character ... Equivalently, K is a finite abelian extension of the field of rational ... ...
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Suppose G is a finite group whose cyclotomic splitting field is a cyclic extension of the rationals. Then, the following numbers are equal: ... ...
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Given a group G, a field k, and a subring R of k, the representation ... function. In the situation of a character-determining field (viz a situation ... ...
1 KB (222 words) - 00:08, 8 May 2008
Suppose G is a group and k is a fact about::class-determining field ... the order of the group is a character-determining field, and hence a class ... ...
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Suppose G is a finite group and K is a field whose characteristic ... * Cyclic implies every field is class-determining
* Elementary abelian ... ...
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group of prime order. Let K be the prime field of characteristic p, i.e ... Then, K is not a class-determining field for G. Specifically, it is ... ...
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Let G be a finite group and k a field. We say that k is sufficiently ... * Character-determining field
* Character-separating field ... ...
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A field is termed class-separating for a group if it satisfies the ... * Character-determining field
==Facts==
For a finite group, a sufficiently ... ...
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A linear character of a group over a field is a homomorphism from the group to the multiplicative group of the field. Equivalently, it can be ... ...
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Let G be a finite group and k a field whose characteristic does not ... splitting: If k is a sufficiently large field for G, then k is a splitting ... ...
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or a prime not dividing the order of G. The field generated by character values ... * In characteristic zero, field generated by character values is contained ... ...
1 KB (169 words) - 00:55, 1 February 2013
Given a group G, a field k, and a subring R of k, the character ring ... by 1, which is either \Z or a prime field of the same characteristic ... ...
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#redirect Field generated by character values need not be a splitting field ... ...
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Suppose G is a finite group and k is a splitting field for G. Then ... Instead of requiring k to be a splitting field, we can require only ... ...
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Note that for a finite group and a field whose characteristic does ... mathbb{Z} which we denote as C_p. Let the field be the prime field on p elements ... ...
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Suppose G is a group and k is a field whose characteristic does not ... # k contains a subfield K that is isomorphic to the field generated ... ...
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Let G be a group, and K be a field.
Then the map \rho: G \to GL_1(K) sending each element of G to the 1 \times 1 identity matrix is a linear representation ... ...
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It is possible to have a finite group such that the fact about::field ... * Field generated by character values is splitting field implies it ... ...
1 KB (145 words) - 20:30, 13 July 2011
we can arrange our example so that the field generated by character values ... character table-equivalent to either. The field generated by character values ... ...
1 KB (211 words) - 03:41, 9 February 2013
Two linear representations of a group over a field are said to be ... Let G be a group and k a field.
Two linear representations \rho_1:G ... ...
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linear representations of G over a field K. Denote by \chi_\alpha, \chi_\beta respectively the characters of \alpha and \beta. Denote by \alpha ... ...
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| degrees of irreducible representations over a splitting field | ... representations over a splitting field || 1,1,1,1,1,1,1,1,1,1,1 ... ...
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