SECTION 4.2 Basics of Group Theory 231
(a) Show that GL 2 (Zp) is a group.
(b) Show that there arep(p+ 1)(p−1)^2 elements in this group.^12
(c) LetBbe the set ofupper triangularmatrices inside GL 2 (Zp).
Therefore,B =
a b
0 c
∣∣
∣∣
∣ac^6 = 0
⊆GL^2 (Zp).Show thatBis a subgroup of GL 2 (Zp), and show that|B|=
p(p−1)^2.
(d) DefineU ⊆B to consist of matrices with 1s on the diagonal.
Show thatU is a subgroup ofBand consists ofpelements.4.2.7 Lagrange’s theorem
In this subsection we shall show a potentially surprising fact, namely
that ifHis a subgroup of the finite groupG, then the order|H|evenly
divides the order|G|ofG. This severly restricts the nature of subgroups
ofG.
The fundamental idea rests on an equivalence relation in the given
group, relative to a subgroup. This relationship is very similar to
the congruence relation ( modn) on the additive groupZof integers.
Thus, let (G,∗) be a group and letH ⊆ G be a subgroup. Define a
relation onG, denoted ( modH) defined by stipulating that
g≡g′( modH) ⇔g−^1 g′∈H.This is easy to show is an equivalence relation:
(^12) This takes a little work. However, notice that a matrix of the form
a b
c d
will have nonzero
determinant precisely when not bothaandbare 0 and when the “vector” (c,d) is not a multiple
of the “vector” (a,b). This implies that there arep^2 −1 possibilities for the first row of the matrix
andp^2 −ppossibilities for the second row. Now put this together!