Download Math 611 HW 4: Due Tuesday, April 6th 1. Let n be a positive integer

Math 611
HW 4: Due Tuesday, April 6th
1. Let n be a positive integer, let F be a field, and suppose α ∈ F. For i, j
with 1 ≤ i, j ≤ n, define the matrix Xij (α) to be the matrix with xkk = 1
for all k, 1 ≤ k ≤ n, xij = α, and all other entries are zero. Prove that
Xij (α) has determinant one.
2. If n is a positive integer and F is a field, show that SL(n, F) is a normal
subgroup of GL(n, F). Also, if F has q elements, show that
n
n2 −n Y
|SL(n, F)| = q 2
(q i − 1).
i=2
3. Prove that the center of SL(n, F) is the set of all diagonal matrices cI
such that cn = 1 in F. (This is just like we did in class with GL(n, F).)
4. Prove that P SL(2, 3) is isomorphic to A4 as follows:
(a) Show that SL(2, 3) has order 24 and has 4 Sylow 3-subgroups (You can
probably do this by counting elements of order 3).
(b) Show that if SL(2, 3) acts by conjugation on the set of Sylow 3-subgroups,
then the kernel of this action is {I, −I} = Z(SL(2, 3)). Therefore there is
an isomorphism of SL(2, 3) to a subgroup of S4 .
(c) Show that A4 is the only subgroup of order 12 in S4 , and thus A4 must
be the subgroup in part (b).
5. Let Xi,j (α) be a transvection. Prove:
(a) Xi,j (α) is in the Borel subgroup B if and only if i < j.
(b) Xi,j (α)Xi,j (β) = Xi,j (α + β), and thus the set {Xi,j (α)|α ∈ F} forms a
subgroup of SL(n, F) isomorphic to F.
(c) Suppose k 6= j. Then Xi,j (α)ek = ek , where ek is the kth standard
column vector (i.e. ek has 1 in the kth row and zero elsewhere).
(d) Xi,j (α)ej = ej + αei .
(e) If g ∈ GL(n, F), then the ith row of Xi,j (α)g is the sum of the ith row of
g and α times the jth row of g. Moreover, the kth row (if k 6= j) of Xi,j (α)g
is the kth row of g.
6. Let Mi,j = Xj,i (1)Xi,j (−1)Xj,i (1). Show that Mi,j ei = ej , Mi,j ej = −ei ,
and Mi,j ek = ek for k 6= i, j, where here the el are again the standard column
basis vectors.
7. Suppose i, j, and k are distinct. Show that [Xi,j (α), Xj,k (β)] = Xi,k (αβ).
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GAP problem
Problem Use GAP to make a conjecture about the size of the center of
SL( n, q), where q is a power of a prime.