Three subgroup lemma

The three-subgroup lemma is a statement from the mathematical branch of group theory . It is a direct consequence of Witt's identity , also known as the Hall-Witt identity , which is named after Ernst Witt and Philip Hall .

Definitions

It is a group . Well known means ${\displaystyle G}$

${\displaystyle [a,b]:=a^{-1}b^{-1}ab}$ For ${\displaystyle a,b\in G}$

the commutator of and . Higher commutators are then defined inductively ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle [a_{1},\ldots ,a_{n}]:=[[a_{1},\ldots ,a_{n-1}],a_{n}]}$ For ${\displaystyle n\geq 3,a_{1},\ldots ,\,a_{n}\in G}$

Are sub-groups , it should be by all commutators , subgroup generated . For subgroups one then explains inductively ${\displaystyle A,B\subset G}$ ${\displaystyle [A,B]}$ ${\displaystyle [a,b],\,a\in A,b\in B}$ ${\displaystyle A_{1},\ldots ,A_{n}\subset G}$

${\displaystyle [A_{1},\ldots ,A_{n}]:=[[A_{1},\ldots ,A_{n-1}],A_{n}]}$.

Note that the set of commutators generally does not form a subgroup and that this inductive definition therefore has to be passed repeatedly to the subgroup created.

Finally, we recall the definition of conjugation . Is , it is a automorphism on which one likes to write as a power: ${\displaystyle x\in G}$ ${\displaystyle a\mapsto xax^{-1}}$ ${\displaystyle G}$

${\displaystyle a^{x}:=x^{-1}ax}$for . ${\displaystyle a,x\in G}$

Witt identity

It is a group. Then Witt's identity applies to everyone ${\displaystyle G}$ ${\displaystyle a,b,c\in G}$

${\displaystyle [a,b^{-1},c]^{b}\,[b,c^{-1},a]^{c}\,[c,a^{-1},b]^{a}=1}$ for all ${\displaystyle a,b,c\in G}$

where denotes the neutral element of the group. ${\displaystyle 1}$

In order to be able to memorize this identity better, note that the exponent always coincides with the middle inverted element, and that the second and third factors result from the first through cyclical exchange . ${\displaystyle [a,b^{-1},c]^{b}}$ ${\displaystyle a\to b\to c\to a}$

This identity is also called the Hall-Witt identity. Philip Hall attributed this equation to Ernst Witt, but the latter was not aware of it. This identity can also be found in the following form:

${\displaystyle [a,b,c^{a}]\,[c,a,b^{c}]\,[b,c,a^{b}]=1}$for all . ${\displaystyle a,b,c\in G}$

If one observes that the conjugation with is an automorphism, the reverse of which is the conjugation with , then the calculation shows ${\displaystyle a}$ ${\displaystyle a^{-1}}$

${\displaystyle [a,b,c^{a}]=[[a,b],c^{a}]=[[a,b]^{a^{-1}},c]^{a}=[[b,a^{-1}],c]^{a}=[b,a^{-1},c]^{a}}$,

that this is actually a variant of Witt's identity.

proof

The proof of Witt's identity is nothing more than a simple calculation after writing out the definitions:

${\displaystyle [a,b^{-1},c]^{b}[b,c^{-1},a]^{c}[c,a^{-1},b]^{a}}$

${\displaystyle =b^{-1}[a^{-1}bab^{-1},c]b\,\cdot \,c^{-1}[b^{-1}cbc^{-1},a]c\,\cdot \,a^{-1}[c^{-1}aca^{-1},b]a}$

${\displaystyle =b^{-1}b{\color {Blue}a^{-1}b^{-1}ac^{-1}a^{-1}}{\color {Red}bab^{-1}cb}\,\cdot \,c^{-1}c{\color {Red}b^{-1}c^{-1}ba^{-1}b^{-1}}{\color {Green}cbc^{-1}ac}\,\cdot \,a^{-1}a{\color {Green}c^{-1}a^{-1}cb^{-1}c^{-1}}{\color {Blue}aca^{-1}ba}}$

${\displaystyle =1}$,

Formula parts of the same color cancel each other out, first the black parts of the formula, then red and green and finally blue.

Note on the definitions

The definitions of commutators and conjugation are not uniform in the literature. Alternatively, if one defines for elements of a group : ${\displaystyle a,b,a_{1},\ldots ,a_{n},x}$ ${\displaystyle G}$

${\displaystyle [a,b]=aba^{-1}b^{-1}}$

${\displaystyle [a_{1},\ldots ,a_{n}]:=[a_{1},[a_{2},\ldots ,a_{n}]]}$

${\displaystyle a^{x}:=xax^{-1}}$,

so the Wittian identity also applies to these definitions.

The three subgroup lemma

• If there are subgroups of a group and is and , then also applies . ${\displaystyle A,B,C}$ ${\displaystyle [A,B,C]=\{1\}}$ ${\displaystyle [B,C,A]=\{1\}}$ ${\displaystyle [C,B,A]=\{1\}}$

Namely , , , it follows by assumption and therefore after Witt's identity, because the conjugation is an automorphism mapping and must be 1 to 1. So everyone interchanges with everyone and therefore with the group created by it , and it follows from this . ${\displaystyle a\in A}$ ${\displaystyle b\in B}$ ${\displaystyle c\in C}$ ${\displaystyle [a,b,c]=[b,c,a]=1}$ ${\displaystyle [c,b,a]=1}$ ${\displaystyle a\in A}$ ${\displaystyle [c,b]}$ ${\displaystyle [C,B]}$ ${\displaystyle [C,B,A]=\{1\}}$

The following statement, also known as the three-subgroup lemma, applies somewhat more generally:

• If subgroups and are a normal divisor of a group and is and , then also applies . ${\displaystyle A,B,C}$ ${\displaystyle N}$ ${\displaystyle [A,B,C]\subset N}$ ${\displaystyle [B,C,A]\subset N}$ ${\displaystyle [C,A,B]\subset N}$

Bertram Huppert attributes this lemma to Philip Hall . It is clear that the special case leads to the first form of the three-subgroup lemma. ${\displaystyle N=\{1\}}$

Applications

Nilpotent groups

One defines inductive for a group ${\displaystyle G}$

${\displaystyle G^{1}=G,\quad G^{n+1}=[G,G^{n}]}$

and calls a group nilpotent if there is one with . An important lemma is ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle G^{n}=\{1\}}$

${\displaystyle [G^{m},G^{n}]\subset G^{m+n}}$for everyone . ${\displaystyle m,n\in N}$

The three-subgroup lemma (in its simpler form) can be used for the induction proof of this lemma.

Abelian groups

Let it be a group and subgroups. Is called ${\displaystyle G}$ ${\displaystyle A,B}$

${\displaystyle N_{G}(B):=\{x\in G|\,x^{-1}Bx=B\}}$the normalizer of in and ${\displaystyle B}$ ${\displaystyle G}$

${\displaystyle C_{A}(B):=\{a\in A|\,[a,b]=1\,\forall b\in B\}}$the centralizer of in . ${\displaystyle B}$ ${\displaystyle A}$

• Is now and , then is Abelian. ${\displaystyle A\subset N_{G}(B)}$ ${\displaystyle A\subset C_{G}([A,B])}$ ${\displaystyle A/C_{A}(B)}$

There is a normal divider and it is to be shown that is . Since, according to the assumption , it follows and because of that , the assertion now follows from the three-subgroup lemma. ${\displaystyle A\subset N_{G}(B)}$ ${\displaystyle C_{A}(B)\subset A}$ ${\displaystyle [A,A]\subset C_{A}(B)}$ ${\displaystyle [A,A,B]=\{1\}}$ ${\displaystyle A\subset C_{G}([A,B])}$ ${\displaystyle [A,B,A]=\{1\}}$ ${\displaystyle [A,B]=[B,A]}$ ${\displaystyle [B,A,A]=\{1\}}$

Individual evidence

1. ↑ Gernot Stroth: Finite Groups, An Introduction , Walter de Gruyter - Verlag (2013), ISBN 978-3-11-029157-5 , Lemma 1.2.5
2. ↑ B. Huppert: Endliche Gruppen I , Springer-Verlag (1967), Chapter III, § 1, Sentence 1.4
3. ↑ Yakov Berkovich, Zvonimir Janko: Groups of Prime Power Order. Volume 1 , Verlag Walter de Gruyter GmbH & Co.KG (2008), ISBN 978-3-11-020822-1 , Introduction, Exercise 13
4. ↑ Steven Roman: Fundamentals of Group Theory An Advanced Approach , Birkhäuser Boston (2011), ISBN 978-0-8176-8301-6 , Theorem 3.43
5. ↑ BAF Wehrfritz: Finite Groups , World Scientific Publishing Co. Pte. Ltd. (1999), ISBN 981-02-3874-6 , page 11
6. ^ Charles Richard Leedham-Green, Susan McKay: The Structure of Groups of Prime Power Order , Oxford University Press (2002), ISBN 0-19-853548-1 , sentence 1.1.6
7. ↑ Hans Kurzweil, Bernd Stellmacher: Theory of finite groups , Springer-Verlag (1998), ISBN 3-540-60331-X , section 1.5.6
8. ↑ Gernot Stroth: Finite Groups, An Introduction , Walter de Gruyter - Verlag (2013), ISBN 978-3-11-029157-5 , Lemma 1.2.6
9. ↑ Steven Roman: Fundamentals of Group Theory An Advanced Approach , Birkhäuser Boston (2011), ISBN 978-0-8176-8301-6 , Corollary 3.44
10. ↑ B. Huppert: Endliche Gruppen I , Springer-Verlag (1967), Chapter III, § 1, Proposition 1.10 (b)
11. ↑ Ernest Shult, David Surowski: Algebra A Teaching and Source Book , Springer-Verlag (2015), ISBN 978-3-319-19733-3 , Corollary 5.2.2