230 CHAPTER 4 Abstract Algebra
- Show that the even integers 2Zis a subgroup of the additive group
of the integers (Z,+). In fact, show that ifnis any positive integer,
then the setnZof multiples ofnis a subgroup of (Z,+). - Show that any subgroup H of the additive group (Z,+) of the
integers must be cyclic. - Show that any subgroup H 6 ={ 0 }of the additive group (C,+) of
complex numbers must be infinite. - Consider the group G = GL 2 (R) of 2×2 matrices of non-zero
determinant. Find an element (i.e., a matrix) A of finite order
and an element B of infinite order. Conclude that G has both
finite and infinite subgroups. - LetX={ 1 , 2 , 3 , 4 }and setG= Sym(X), the group of permuta-
tions ofX. Find all of the elements inGhaving order 2. Find all
of the elements ofGhaving order 3. Find all of the elements ofG
having order 4. - Let (G,∗) be a cyclic group and let H ⊆G, H 6 ={e} be a sub-
group. Show that H is also cyclic. (This is not entirely trivial!
Here’s a hint as to how to proceed. LetG have generatorxand
letn be thesmallest positive integer such that xn ∈H. Show
that, in fact,xn is a generator ofH.) - Consider the setR+of positive real numbers and note that (R+,·)
is a group, where “·” denotes ordinary multiplication. Show that
R+has elements of finite order as well as elements of infinite order
and hence has both finite and infinite subgroups. - Consider a graph with set X of vertices, and letG be the auto-
morphism group of this graph. Now fix a vertexx ∈X and set
Gx={σ∈G|σ(x) =x}. Prove thatGxis a subgroup ofG, often
called thestabilizerinGof the vertexx. - Find the orders of each of the elements in the cyclic group (Z 12 ,+).
- Letpbe a prime number and letZpbe the integers modulo three
and consider the group GL 2 (Zp) of matrices having entries inZ 3
and all having nonzero determinant.