Group Theory Properties @grouptheory_bot

Let G be a group. Let p be a prime. #math #mathematics #algebra #grouptheory #science Joined November 2018

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Group Theory Properties @grouptheory_bot

2 years ago

Let K be a characteristic subgroup of N. And let N be a normal subgroup of G. Then, K is a normal subgroup of G.

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Group Theory Properties @grouptheory_bot

2 years ago

Let G be field, and ρ be an irreducible representation over the field of complex numbers. Then |G| is devided by the degree of ρ.

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Group Theory Properties @grouptheory_bot

2 years ago

【Burnside's theorem】Let P be a Sylow p-subgroup of G. Suppose that P is contained in the center of the normalizer of P. Then, G has a normal p-complement.

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Group Theory Properties @grouptheory_bot

2 years ago

Let Sylp(G) be the set of Sylow p-subgroups of G. And let S be a Sylow p-subgroup. Then, the following are equivalent. (1) S is normal of G. (2) Sylp(G)={S}

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Group Theory Properties @grouptheory_bot

2 years ago

Let N be an arbitrary normal subgroup such that G/N is abelian. Then, the commutator subgroup G' is contained in N. In other words, G' is the unique normal subgroup of G with the minimal property that the corresponding factor group is abelian.

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Group Theory Properties @grouptheory_bot

2 years ago

【Maschke】Let k be a field, and (ρ, V) be a representation of G over k. And suppose the character of k doesn't divide the order of G. Then, (ρ, V) is completely reducible. In particular, if the character of k is 0, (ρ, V) is completely reducible.

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Group Theory Properties @grouptheory_bot

2 years ago

Let M be the Frattini subgroup of P, where P is a p-group. And let N be normal of P. P/N is elementary abelian if and only if M is contained in N.

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Group Theory Properties @grouptheory_bot

2 years ago

Let Sylp(G) be the set of Sylow p-subgroups of G. And let S be a Sylow p-subgroup. Then, the following are equivalent. (1) S is normal of G. (2) Sylp(G)={S}

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Group Theory Properties @grouptheory_bot

2 years ago

【Dedekind's lemma】Let H, K, L be subgroups of G. And assume H is contained in L. Then, H(K∩L)=HK∩L

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Group Theory Properties @grouptheory_bot

2 years ago

Let G be field, and ρ be an irreducible representation over the field of complex numbers. Then |G| is devided by the degree of ρ.

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Group Theory Properties @grouptheory_bot

2 years ago

【Schur-Zassenhaus theorem】Let G be finite, and N be a normal subgroup of G. And assume (|G|, |G:N|)=1. Then, N is complemented in G.

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Group Theory Properties @grouptheory_bot

2 years ago

【Burnside】Let G be finite. And let Χ be an irreducible character of G with degΧ is not equal to 1. Then, there exists x in G such that Χ(x)=0.

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Group Theory Properties @grouptheory_bot

2 years ago

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Group Theory Properties @grouptheory_bot

2 years ago

Let G be finite, and Χ be an irreducible character of G. And suppose x in K, where K is a conjugacy class of G. Then, |K|Χ(x)/Χ(1) is an algebraic integer.