2
K. W . GRUENBER G
F
F-
771
c
*
^ 2
—±-G
situation wa s initiate d b y B . H . Neuma n i n hi s thesi s i n th e mid-thirties .
(Cf. [33] , [11], [12].)
But tw o presentation s ar e rarel y linke d s o strongly . A much weake r
connexion woul d b e th e numbe r o f th e definin g relations . 6
Question 1. Given 1 —»/?.—* F J G —• 1 /o r z = 1, 2 , z' s //?e «
The answe r i s unknow n fo r finit e group s G . I t i s definitel y ''no " fo r th e wide r
class o f finitel y presentabl e groups : th e grou p (a, b; a - b ) als o ha s a 2-generato r
presentation needin g mor e tha n on e relation . (Cf . [14]; als o [4] , [13].)
Suppose nex t tha t w e allo w F t o vary . Le t r(G ) b e th e minimu m o f th e number s o f
relations neede d t o defin e G . W e as k
Question 2 . Is r(G) realised in a minimal presentation (i.e., one in which d(F) -
d{G))?
This to o i s unknow n (fo r finit e groups) . Bot h question s woul d hav e a n affirmativ e
answer i f th e followin g wer e true :
Question 3 . Is d
F
(R) - d(F) independent of the presentation (1) and therefore an
invariant for G ?
Note that , o f course , thi s fail s fo r th e one-relato r grou p give n above .
I .4 An y extensio n
(2) 1 K - E - ^ G 1
is a n imag e o f a suitabl e fre e presentatio n (1).W e simpl y tak e a fre e presentatio n f
of E an d se t n - f6. Then Rf ~ K.
l'— R -+ F - ^ G 1
1*1. 1
1 K~ E G 1.
In thi s way , presentatio n theor y ca n b e use d t o stud y th e possibl e extension s ove r G .
It i s immediate , fo r example , tha t d
£
(K) d
p
{R).
In man y situation s K i s restricte d i n som e way . Fo r instance , K migh t hav e t o b e
finite, o r soluble , o r centra l i n E. W e may expec t tha t the n presentatio n theor y wil l
provide correspondingl y mor e information .
1 . 5 Suppos e K i s centra l i n E . The n Ke r 4 [R, F] an d (1) shoul d b e replace d
by th e free central extension
1 R/[R, F) F/[R, F] - G 1.
(This terminolog y ca n b e justifie d i n a rigorou s manner : cf . [18, §9.9].)
Here Questio n 3 has a n affirmativ e answer : d(R/[R, F]) - d{F) is a n invarian t fo r
G. Bu t somethin g muc h bette r i s true .
Since R/R O F* ^ RF'/F' an d the right hand side i s fre e abelia n o f ran k d{F) (recal l
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