Find the elementary divisors and invariant factors of .
We decompose the group (\mathbb{Z}*8 \times \mathbb{Z}*{12} \times \mathbb{Z}*{15}) into its primary components. Writing each modulus as a product of prime powers gives (8=2^3), (12=2^2\cdot 3), and (15=3\cdot 5). Hence the (2)-primary part is (\mathbb{Z}*{2^3}\times \mathbb{Z}_{2^2}), the (3)-primary __________ _____ _________ ______ ______ _____ __________ _________ ______ _____ __________ __________.
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Let . Show that G is the cyclic group of order six.
(b) Solve the following set of congruences:
(c) Show that is not a UFD by giving two different factorisations of 20.
Check that is in the stabiliser of
. Also, show that there are infinitely many choices of
for which
is invertible.
Which of the following statements are true and which are false? Give reasons for your answer.
(a) If a finite group G acts on a finite set S, then for all
.
(b) There are exactly 8 elements of order 3 in S4.
(c) If , then
.
(d) .
(e) For any ,
.
In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group
Let H be a finite group and, for some prime p, let P be a p-Sylow subgroup of H which is normal in H. Suppose H is normal in K, where K is a finite group. Then, show that P is normal in K.
Suppose n is even, , where
. Let
and
. Show that HN is a subgroup of Dn. What is its order?
If , show that a polynomial ax2 + bx + c is irreducible iff
where
is the group of squares in