Some solutions to problems from Chapter 5

6. Show that A_{8 }contains an element of order 15.

The product of a 5-cycle and a 3-cycle will belong to A_{8
}and will have order 15.

(1 2 3 4 5) (6 7 8) will do.

8. What is the maximum order of any element of A_{10}?

We can see that (1 2 3 4 5 6 7) (8 9 10) has order 21. The element (1 2 3 4 5)(6
7 8)(9 10) has order 30, but is odd, so not in A_{10}. It would be best
to show all orders.

10. Prove that a function from a finite set S to itself is 1-1 if and only if it is onto.

Suppose |*S*| = n,
and let _{} be a function. If the function is 1-1, then _{} is a subset of *S* with n elements, so must be all
of *S*,
and the function is onto. Conversely, if
the function is onto, then, if it fails to be 1-1, there are *a*, *b*
in *S*, with _{}. Then _{} can have no more than *n-1* elements, but has the same image as *S*, so the function cannot be onto.

The result may fail for infinite sets. The function sending the natural numbers to itself by mapping n to n + 1, for each n, is 1-1 but not onto, for example.

12. Prove: If a permutation is even then its inverse is even; if a permutation is odd, its inverse is odd.

Let α be even.
Then it is the product of *r*
transpositions, for some even *r*
. By the s-s lemma, its inverse may be
written as a product of the same transpositions, in the opposite order, and
must be even. The same statement, with
even changed to odd, proves the rest of the assertion

A prettier argument says, let α and its inverse be products of *r* and *s* transpositions, respectively, then the product of the two gives a
representation of the identity as a product of *r + s* transpositions, so *r +
s* must be even, by the lemma in the book.
Then *r* and *s* must both be even or both be odd.

28. Let β = (123) (145). Write β^{99} in disjoint cycle
form.

First, β = (14523).
Then β^{99} = (β^{5})^{19}β^{4}
= β^{4} = (13254)

36. In *S*_{4},
find a cyclic subgroup of order 4 and a non-cyclic subgroup of order 4.

{(1234), (1234)^{2} = (13) (24), (1234)^{3} =
(1432), (1234)^{4} = (1)} is a cyclic subgroup of order 4.

{(12), (34), (12)(34), (1)} is a non-cyclic subgroup of order 4.

46. Suppose that *H*
is a subgroup of *S*_{n}
of odd order. Prove that *H* is a subgroup of *A*_{n}.

This result follows quickly from problem 19. I shall prove it using the argument for
problem 19 on the way. If a subgroup of *S*_{n}_{
}is not a subgroup of *A*_{n}, then it has both even and odd permutations—the
identity is even, of course. But then,
if we let α be an odd permutation, the function λ_{α},
as defined in the proof of the Cayley representation
theorem—just using this to set notation; we already knew about “translation
functions”—gives a 1-1 mapping from the odd permutations in *H* onto the even permutations in *H*.
Thus, exactly half of the elements of *H* are even. Since *H* is of odd order, this can’t happen, so
*H* is a subgroup of *A*_{n}.