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2.54 (i) Deﬁne the special linear group by
SL(2, R) = { A ∈ GL(2, R) : det( A) = 1}.
Prove that SL(2, R) is a subgroup of GL(2, R).
Solution. It sufﬁces to show that if A , B ∈ SL(2, R), then so is
AB −1 ; that is, if det( A) = 1 = det( B ), then det( AB −1 ) = 1.
Since det(U V ) = det(U ) det(V ), it follows that
1 = det( E ) = det( B B −1 ) = det( B ) det( B −1 ).
Hence, det( AB −1 ) = det( A) det( B −1 ) = 1. (ii) Prove that GL(2, Q) is a subgroup of GL(2, R).
Solution. Both the product of two matrices with rational entries
and the inverse of a matrix with rational entries have rational entries.
2.55 Give an example of two subgroups H and K of a group G whose union
H ∪ K is not a subgroup of G .
Solution. If G = S3 , H = (1 2) , and K = (1 3) , then H ∪ K is not a
subgroup of G , for (1 2)(1 3) = (1 3 2) ∈ H ∪ K .
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2.56 Let G be a ﬁnite group with subgroups H and K . If H ≤ K , prove that
[G : H ] = [G : K ][ K : H ].
Solution. If G is a ﬁnite group with subgroup H , then [G : H ] = G / H .
Hence, if H ≤ K , then
[G : K ][ K : H ] = (G / K ) · ( K / H ) = G / H  = [G : H ].
2.57 If H and K are subgroups of a group G and if  H  and  K  are relatively
prime, prove that H ∩ K = {1}.
Solution. By Lagrange’s theorem,  H ∩ K  is a divisor of  H  and a divisor
of  K ; that is,  H ∩ K  is a common divisor of  H  and  K . But  H  and
 K  are relatively prime, so that  H ∩ K  = 1 and H ∩ K = {1}.
2.58 Prove that every inﬁnite group contains inﬁnitely many subgroups.
Solution. An inﬁnite group have only ﬁnitely many cyclic subgroups.
2.59 Let G be a group of order 4. Prove that either G is cyclic or x 2 = 1 for
every x ∈ G . Conclude, using Exercise 2.44, that G must be abelian.
Solution. If G has order 4, then the only possible orders of elements in G
are 1, 2, and 4. If there is an element of order 4, then G is cyclic with that
element as generator. Otherwise, every element has order 1 or 2, so that
x 2 = 1 for every x ∈ G . ...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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