Homework 5, due 9/25/2015
1. Drawing from groups we have already encountered in class, show that the
property of being a normal subgroup is not transitive, i.e., there exists a
group G and subgroups K E H E G with K normal in H, and H normal
in G, but K not normal in G.
√
2. (Artin) Prove that the numbers of the form a + b 2, where a and b are
rational numbers, form a subfield of C.
3. (Artin) Solve completely the
AX = B, where

1 1
A = 1 0
1 −1
systems of linear equations AX = 0 and

0
1 ,
−1


1
and B = −1 ,
1
(a) in Q;
(b) in F2 ;
(c) in F3 ;
(d) in F7 .
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4. (Artin+supplement) Determine the order of the matrices
0
2 0
in the group GL2 (F7 ). Also, for all primes p, exhibit an
0 1
of order p in GL2 (Fp ).
1
1
and
element
∼
5. We saw that there is an isomorphism f : GL2 (F2 ) −
→ S3 (see also Artin
3.2.9). Compute the composite map
f
ε
GL2 (F2 ) −
→ S3 −
→ {±1},
where ε denotes the sign homomorphism (i.e. the determinant of the
associated permutation matrix), Of course, since S3 has non-trivial automorphisms, the choice of isomorphism f is not unique; nevertheless, show
that the composite ε ◦ f is independent of the choice of isomorphism f .
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