MATH 516 Hour Exam Radford 10/13/06 10AM
Name (print)
(1) Return this exam copy with your exam booklet. (2) Write your solutions in your
exam booklet. (3) Show your work. (4) There are five questions on this exam. (5) Each
question counts 20 points. (6) You are expected to abide by the University’s rules concerning
academic honesty.
1. Let G=<a> be a cyclic group of order 22.
(a) Find the number of subgroups of G.
(b) Find |a−94|.
(c) List the generators of Gin the form a`, where 0 ≤` < 22.
(d) List the elements of <a46>in the form a`, where 0 ≤` < 22.
(e) Draw the lattice diagram for G.
2. Let GL2(R) be the group of invertible 2×2 matrices with real coefficients under matrix
multiplication and set
G={Ãa0
0b!|a, b ∈Rand ab 6= 0}.
(a) Show that G≤GL2(R).
(b) The group Gacts on A=R2on the left by matrix multiplication. Describe the
G-orbits of A. How many are there?
3. Let f , g :G−→ G0be homomorphisms.
(a) Show that
H={a∈G|f(a) = g(a)}
is a subgroup of G.
(b) Let a∈G. Show, by induction, that f(an) = f(a)nfor all n≥0. (Use the definition
a0=eand an+1 =aanfor n≥0.)
4. Let Gbe a finite group and suppose that H, K ≤G.
(a) Suppose that H, K £G. Show that HK £G.
(b) Suppose that |H|=`and His the only subgroup of Gof order `. Show that H£G.
5. Let Gbe a finite group of order 3·5·19.
(a) Show that Ghas a normal subgroup of order 19.
(b) Show that Ghas a subgroup of index 3.
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