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Proposition 3.8.1.
Let H be a subgroup of the group G,
and let a,b be elements of G.
Then the following conditions are equivalent:
A result similar to Proposition 3.8.1 holds for right cosets.
Let H be a subgroup of the group G,
and let a,b belong to G.
Then the following conditions are equivalent:
Definition 3.8.2.
Let H be a subgroup of the group G, and let a be an element of G.
The set
aH = { x in G | x = ah for some h in H }
is called the left coset of H in G determined by a. Similarly, the right coset of H in G determined by a is the setHa = { x in G | x = ha for some h in H }.
The number of left cosets of H in G is called the index of H in G, and is denoted by [G:H].
Proposition 3.8.3.
Let N be a normal subgroup of G, and let
a,b,c,d be elements of G.
If aN = cN and bN = dN, then abN = cdN.
Theorem 3.8.4.
If N is a normal subgroup of G, then the set of left cosets
of N forms a group under the coset multiplication given by
(aN)(bN) = abN
for all a,b in G.
Definition 3.8.5.
If N is a normal subgroup of G,
then the group of left cosets of N in G is called the
factor group
of G determined by N.
It will be denoted by G/N.
Example 3.8.5.
Let N be a normal subgroup of G.
If a is any element of G,
then the order of aN in G/N is the smallest positive integer n such that
a^{n} belongs to N.
Proposition 3.8.6.
Let N be a normal subgroup of G.
Proposition 3.8.7.
Let H be a subgroup of the group G.
The following conditions are equivalent:
Example 3.8.7.
Any subgroup of index 2 is normal.
Example 3.8.8.
If m is a divisor of n, then
Z_{n} / mZ_{n}
Z_{m}.
3.8.8. Theorem [Fundamental Homomorphism Theorem]
Let G_{1},
G_{2} be groups.
If
: G_{1} -> G_{2}
is a group homomorphism with
K = ker(), then
G_{1} / K (G_{1}).
3.8.9. Definition
The group G is called a
simple
group if it has no proper nontrivial normal subgroups.
In actually using the Fundamental Homomorphism Theorem, it is important to let the theorem do its job, so that it does as much of the hard work as possible. Quite often we need to show that a factor group G/N that we have constructed is isomorphic to another group G_{1}. The easiest way to do this is to just define a homomorphism µ from G to G_{1}, making sure that N is the kernel of µ. If you prove that µ maps G onto G_{1}, then the Fundamental Theorem does the rest of the work, showing that there exists a well-defined isomorphism between the factor group G/N and G_{1}.
The moral of this story is that if you define a function on G rather than G/N, you ordinarily don't need to worry that it is well-defined. On the other hand, if you define a function on the cosets in G/N, the most convenient way is use a formula defined on representatives of the cosets of N. But then you must be careful to prove that the formula you are using does not depend on the particular choice of a representative. That is, you must prove that your formula actually defines a function. Then you must prove that your function is one-to-one, in addition to proving that it is onto and respects the operations in the two groups. Once again, if your function is defined on cosets, it can be much trickier to prove that it is one-to-one than to simply compute the kernel of a homomorphism defined on G.
27. List the cosets of < 7 > in Z_{16}^{×}. Is the factor group Z_{16}^{×} / < 7 > cyclic? Solution
28.
Let G = Z_{6} × Z_{4},
let H = { (0,0), (0,2) }, and let K = { (0,0), (3,0) }.
(a) List all cosets of H; list all cosets of K.
(b) You may assume that any abelian group of order 12 is isomorphic to either
Z_{12} or Z_{6} × Z_{2}.
Which answer is correct for G/H? For G/K?
Solution
29.
Let the dihedral group D_{n}
be given via generators and relations,
with generators a of order n and b of order 2,
satisfying ba=a^{-1} b.
(a) Show that ba^{i} = a^{-i} b for all i with
1 i < n.
(b) Show that any element of the form a^{i} b has order 2.
(c) List all left cosets and all right cosets of < b >
Solution
30.
Let G = D_{6} and let N be the subgroup
< a^{3} > = {e, a^{3} } of G.
(a) Show that N is a normal subgroup of G.
(b) Is G/N abelian?
Solution
31.
Let G be the dihedral group D_{12},
and let N = { e,a^{3},a^{6},a^{9} }.
(a) Prove that N is a normal subgroup of G, and list all cosets of N.
(b) You may assume that G/N is isomorphic to either
Z_{6} or S_{3}.
Which is correct?
Solution
32.
(a) Let G be a group.
For a,b in G we say that b is conjugate to a, written b ~ a,
if there exists g in G such that b = gag^{-1}.
Show that ~ is an equivalence relation on G.
The equivalence classes of ~ are called the
conjugacy classes of G.
(b) Show that a subgroup N of G is normal in G
if and only if N is a union of conjugacy classes.
Solution
33. Find the conjugacy classes of D_{4}. Solution
34.
Let G be a group, and let N and H be subgroups of G
such that N is normal in G.
(a) Prove that HN is a subgroup of G.
(b) Prove that N is a normal subgroup of HN.
(c) Prove that if H N = { e },
then HN / N is isomorphic to H.
Solution
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