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**Definition 3.2.1.**
Let G be a group,
and let H be a subset of G.
Then H is called a
**subgroup**
of G if H is itself a group,
under the operation induced by G.

**Example** 3.2.1.
**Q**^{×} and
**R**^{×} are subgroups of
**C**^{×}, the multiplicative group of complex numbers.

**Example** 3.2.2.
SL_{n}(**R**),
the set of all n × n matrices over **R** with determinant 1,
is a subgroup of
GL_{n}(**R**).

**Proposition 3.2.2.**
Let G be a group with identity element e,
and let H be a subset of G.
Then H is a subgroup of G if and only if
the following conditions hold:

**(i)**for all a,b in H, the product ab is in H;**(ii)**e belongs to H;**(iii)**for all a in H, the inverse a^{-1}is in H.

**Corollary 3.2.3.**
Let G be a group and let H be a subset of G.
Then H is a subgroup of G if and only if
H is nonempty and ab^{-1} is in H for all a,b in H.

**Corollary 3.2.4.**
Let G be a group, and let H be a finite, nonempty subset of G.
Then H is a subgroup of G if and only if
ab is in H for all a,b in H.

**Definition 3.2.5.**
Let G be a group, and let a be any element of G. The set

<a> =
{ x in G | x = a^{n} for some n in Z }

The group G is called a

**Proposition 3.2.6.**
Let G be a group, and let a be an element of G.

**(a)**The set <a> is a subgroup of G.**(b)**If K is any subgroup of G such that a is an element of K, then <a> K.

**Examples** 3.2.7 - 3.2.9.
**Z** and
**Z**_{n} are cyclic groups, but
**Z**_{n}^{×} may not be cyclic.
(See Corollary 7.5.11.)

**Definition 3.2.7.**
Let a be an element of the group G.
If there exists a positive integer n such that
a^{n} = e,
then a is said to have
**finite order**,
and the smallest such positive integer is called the
**order**
of a, denoted by o(a).

If there does not exist a positive integer n such that
a^{n} = e, then a is said to have
**infinite order**.

**Proposition 3.2.8.**
Let a be an element of the group G.

**(a)**If a has infinite order, and a^{k}= a^{m}for integers k,m, them k=m.**(b)**If a has finite order and k is any integer, then a^{k}= e if and only if o(a) | k.**(c)**If a has finite order o(a)=n, then for all integers k, m, we havea

Furthermore, | <a> | = o(a).^{k}= a^{m}if and only if k m (mod n).

**Lemma 3.2.9.**
Let H be a subgroup of the group G.
For a,b in G define a ~ b if ab^{-1} is in H.
Then ~ is an equivalence relation.

**Theorem 3.2.10 (Lagrange).**
If H is a subgroup of the finite group G,
then the order of H is a divisor of the order of G.

**Corollary 3.2.11.**
Let G be a finite group of order n.

**(a)**For any a in G, o(a) is a divisor of n.**(b)**For any a in G, a^{n}= e.

**Example** 3.2.12.
(Euler's theorem)
Let G be the multiplicative group of congruence classes modulo n.
The order of G is given by
(n),
and so by Corollary 3.2.11, raising any congruence class to the power
(n)
must give the identity element.

**Corollary 3.2.12.**
Any group of prime order is cyclic.

Lagrange's theorem is very important. It tells us that in a finite group the number of elements in any subgroup must be a divisor of the total number of elements in the group. This is a useful fact to know when you are looking for subgroups in a given group.

It is also important to remember that every element a in a group G
defines a subgroup < a >
consisting of all powers (positive and negative) of the element.
This subgroup has o(a) elements,
where o(a) is the order
of a.
If the group is finite,
then you only need to look at positive powers,
since in that case the inverse a^{-1} of any element
can be expressed in the form a^{n}, for some n > 0.

For a group G, the
**centralizer**
C(a) of an element a in G
is defined in Exercise 3.2.14 of the text as

C(a) = { x in G | xa = ax } .

Exercise 3.2.14 shows that C(a) is a subgroup of G that contains <a>.
The
**center**
of the group G, denoted by Z(G), is defined in Exercise 3.2.16 as

Z(G) = { x in G | xg = gx for all g in G } .

Exercise 3.2.16 shows that Z(G) is a subgroup of G.
**23.**
Find all cyclic subgroups of **Z**_{24}^{×}.
*Solution*

**24.**
In **Z**_{20}^{×}, find two subgroups of order 4,
one that is cyclic and one that is not cyclic.
*Solution*

**25.**
(a) Find the cyclic subgroup
of S_{7} generated by
the element (1,2,3)(5,7).

(b) Find a subgroup of S_{7} that contains 12 elements.
You do not have to list all of the elements
if you can explain why there must be 12,
and why they must form a subgroup.

*Solution*

**26.**
In G = **Z**_{21}^{×}, show that

H = { [x]_{21} | x
1 (mod 3) }
and
K = { [x]_{21} | x
1 (mod 7) }

**27.**
Let G be an abelian group, and let n be a fixed positive integer.
Show that

N = { g in G | g = a^{n} for some a in G }

**28.**
Suppose that p is a prime number of the form p = 2^{n} + 1.

(a) Show that in **Z**_{p}^{×}
the order of [2]_{p} is 2n.

(b) Use part (a) to prove that n must be a power of 2.

*Solution*

**29.**
In the multiplicative group **C**^{×} of complex numbers,
find the order of the elements

= and = .

**30.**
Let K be the following subset of GL_{2} (**R**).

K =

Show that K is a subgroup of GL
**31.**
Compute the
centralizer
in GL_{2} ( **R**) of the matrix
.
*Solution*

**32.**
Let G be the subgroup of GL_{2} (**R**)
defined by G =
.

Let A =
and B =
.

Find the centralizers C(A) and C(B),
and show that C(A) C(B) = Z(G),
where Z(G) is the center of G.
*Solution*

**Lab 1.**
For each group G of order 8 on the list of groups given by
** Groups15**,
find the elements of order 2.
Do these elements (together with the identity element)
form a subgroup of G?

**Lab 2.**
For each group G of order n < 16,
find the largest possible order of an element a in G.

**Lab 3.**
Find all groups G of order n < 16
for which there is a divisor m of n,
but no corresponding subgroup of G of order m.
(These are the groups of order 15 or less
for which the converse of
Lagrange's theorem
fails.)

**Lab 4.**
If |G| = n, where n < 16, and p is a *prime* divisor of n,
can you always find a subgroup H of G with |H| = p?
What can you say if you ask the question for prime power?
(You only need to worry about n = 8 and n = 12.)

**Lab 5.**
List the groups G in
** Groups15**
for which the
center

Z(G) = { x in G | xg = gx for all g in G }

contains only the identity element.
**Lab 6.**
For a group G, the
centralizer
of an element a in G is defined as

C(a) = { x in G | xa = ax } .

In the group of order 12 called A

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