174 Bayer^Fluckiger, Kearton ста Wilson^ Decomposition of modules, forms and simple knots

( ii ) If fe SgC U) and h £ Sj^Ç, U) for some central element x of U then g »-> g^ induces a bijection of pointed sets

H , C , GJ ) - ^H , , C , h''GhJh )

where the involution ^ is given by £ = А~^^й. Moreover if G is ^-stable then h^^Gh is ^'Stable,

( iii ) Let N be a normal subgroup of G and let N and G be stable under ~ so that we get an involution, also written ~, on G = G/N, Suppose that fe Sg(", G), then we have an exact sequence of pointed sets

Hi\N ) ^H , C , G , f ) - ^H , { - , GJ )

where the first map is induced by n^-^nf and ^ is given by â=fâf^^, {Note that ЩС, G,f) is, as a non-pointed set, independent of f)

( iv ) If H is a subgroup of G of finite index then

HA\H , f ) is finite ^feUoH^\G,f) is finite \/feU.

Suppose U^VxWand G<V, H<W.

( v ) IfV=Vand W^W then HÇ,GxH)^HÇ,G)xHÇ,H). (vi) If f^W and W^V then H{\ GxЯ) = {[1]}. Proof (i) and (ii) are clear, (iii) By (i) with h^f'^ we have an "isomorphism" of set complexes

H { \N ) - - - - - - > HX, G,f)-----> Л,(^ G,f)

H { \N ) - - - - - - > H{\G) -----> H{\G),

The lower row is easily seen to be exact.

( iv ) Let gl,..., g^ be a left transversal of H in G. If /' e S^, GfG) then the fibre of the mapping

HA\H , f ) - ^H , C . G , f )

which lies over [/I0 îs contained in {[gi/'g,]H}- So <= is clear. On the other hand

OfG^\JÉgJgjH .

и

So H^Ç,G,f) is covered by the images of the НД", Я, g|/gj). This establishes the converse.

( v ) and (vi) are dear.