Editing Index of a subgroup (section)

=== Geometric structure ===
An elementary observation is that one cannot have exactly 2 subgroups of index 2, as the [[Complement (set theory)|complement]] of their [[symmetric difference]] yields a third. This is a simple corollary of the above discussion (namely the projectivization of the vector space structure of the elementary abelian group 
:<math>G/\mathbf{E}^p(G) \cong (\mathbf{Z}/p)^k</math>,

and further, ''G'' does not act on this geometry, nor does it reflect any of the non-abelian structure (in both cases because the quotient is abelian).

However, it is an elementary result, which can be seen concretely as follows: the set of normal subgroups of a given index ''p'' form a [[projective space]], namely the projective space 
:<math>\mathbf{P}(\operatorname{Hom}(G,\mathbf{Z}/p)).</math>

In detail, the space of homomorphisms from ''G'' to the (cyclic) group of order ''p,'' <math>\operatorname{Hom}(G,\mathbf{Z}/p),</math> is a vector space over the [[finite field]] <math>\mathbf{F}_p = \mathbf{Z}/p.</math> A non-trivial such map has as kernel a normal subgroup of index ''p,'' and multiplying the map by an element of <math>(\mathbf{Z}/p)^\times</math> (a non-zero number mod ''p'') does not change the kernel; thus one obtains a map from 
:<math>\mathbf{P}(\operatorname{Hom}(G,\mathbf{Z}/p)) :=  (\operatorname{Hom}(G,\mathbf{Z}/p))\setminus\{0\})/(\mathbf{Z}/p)^\times</math>

to normal index ''p'' subgroups. Conversely, a normal subgroup of index ''p'' determines a non-trivial map to <math>\mathbf{Z}/p</math> up to a choice of "which coset maps to <math>1 \in \mathbf{Z}/p,</math> which shows that this map is a bijection.

As a consequence, the number of normal subgroups of index ''p'' is 
:<math>(p^{k+1}-1)/(p-1)=1+p+\cdots+p^k</math>

for some ''k;'' <math>k=-1</math> corresponds to no normal subgroups of index ''p''. Further, given two distinct normal subgroups of index ''p,'' one obtains a [[projective line]] consisting of <math>p+1</math> such subgroups.

For <math>p=2,</math> the [[symmetric difference]] of two distinct index 2 subgroups (which are necessarily normal) gives the third point on the projective line containing these subgroups, and a group must contain <math>0,1,3,7,15,\ldots</math> index 2 subgroups – it cannot contain exactly 2 or 4 index 2 subgroups, for instance.