Burnside's basis theorem: Difference between revisions

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Statement

Symbolic statement

Let $P$ be a $p$ -group for some prime $p$ , and let $\Phi (P)$ denote the Frattini subgroup of $P$ . Then, $P/\Phi (P)$ is the largest elementary Abelian quotient of $P$ , and hence is a vector space over the prime field $\mathbb {F} _{p}$ .

Burnside's basis theorem states that:

A subset $S$ of $P$ is a generating set for $P$ iff the image of $S$ in $P/\Phi (P)$ generates $P/\Phi (P)$ as a $\mathbb {F} _{p}$ -vector space.
A subset $S$ of $P$ is a minimal generating set for $P$ iff the image of $S$ in $P/\Phi (P)$ is a vector space basis for $P/\Phi (P)$ .

Proof

The proof follows directly from the following two facts:

Combining a generating set for a normal subgroup, and a set of inverse images (via the quotient map) of the quotient group gives us a generating set for the quotient group
Any element in the Frattini subgroup can be dropped from any generating set.

@@ Line 5: / Line 5: @@
 ===Symbolic statement===
-Let <math>P</math> be a <math>p</math>-group for some prime <math>p</math>, and let <matH>\Phi(P)</math> denote the [[Frattini subgroup]] of <math>P</math>. Then, <math>P/\Phi(P)</math> is the largest elementary Abelian quotient of <math>P</math>, and hence is a vector space over the prime field <math>F_p</math>.
+Let <math>P</math> be a <math>p</math>-group for some prime <math>p</math>, and let <math>\Phi(P)</math> denote the [[Frattini subgroup]] of <math>P</math>. Then, <math>P/\Phi(P)</math> is the largest elementary Abelian quotient of <math>P</math>, and hence is a vector space over the prime field <math>\mathbb{F}_p</math>.
-Burnside's basis theorem states that given any vector space basis for <math>P/\Phi(P)</math> and an arbitrary choice of one inverse image of each basis element under the quotient map <math>P \to P/\Phi(P)</math>, the set of inverse images forms a generating set for the whole of <math>P</math>
+Burnside's basis theorem states that:
+* A subset <math>S</math> of <math>P</math> is a [[generating set]] for <math>P</math> iff the image of <math>S</math> in <math>P/\Phi(P)</math> generates <math>P/\Phi(P)</math> as a <math>\mathbb{F}_p</math>-vector space.
+* A subset <math>S</math> of <math>P</math> is a [[minimal generating set]] for <math>P</math> iff the image of <math>S</math> in <math>P/\Phi(P)</math> is a vector space basis for <math>P/\Phi(P)</math>.
 ==Proof==