This gives a total of 4 such symmetries, thus | Sym( R) | = 4.
Again we can describe symmetries in terms of their effect on the vertices. Here are the 4
elements of Sym( R) described in permutation notation.
µ
¶
µ
¶
µ
¶
µ
¶
A B C D
A B C D
A B C D
A B C D
ι = A B C D
B A D C
C D A B
D C B A
Given a regular n-gon ( i.e. , a regular polygon with n sides all of the same length and n
vertices V 1 , V 2 , . . . , Vn) the symmetry group is the dihedral group of order 2 n D 2 n, with elements ι, α, α 2 , . . . , αn− 1 , τ, ατ, α 2 τ, . . . , αn− 1 τ
where αk is an anticlockwise rotation through 2 πk/n about the centre and τ is a reflection in
the line through V 1 and the centre. Moreover we have
|α| = n, |τ | = 2 , τ ατ = αn− 1 = α− 1 .
6. SUBGROUPS AND LAGRANGE’S THEOREM
35
In permutation notation this becomes
α = ( V 1 V 2 · · · Vn) ,
but τ is more complicated to describe.
For example, if n = 6 we have
α = ( V 1 V 2 V 3 V 4 V 5 V 6) ,
τ = ( V 2 V 6)( V 3 V 5) ,
while if n = 7
α = ( V 1 V 2 V 3 V 4 V 5 V 6 V 7) ,
τ = ( V 2 V 7)( V 3 V 6)( V 4 V 5) .
We have seen that when n = 3, Sym( 4) is the permutation group of the vertices and so D 6 is
essentially the same group as S 6.
6. Subgroups and Lagrange’s Theorem
Let ( G, ∗) be a group and H ⊆ G. Then H is a subgroup of G if ( H, ∗) is a group. In detail this means that
• for x, y ∈ H, x ∗ y ∈ H;
• ι ∈ H;
• if z ∈ H then z− 1 ∈ H.
We write H 6 G whenever H is a subgroup of G and H < G if H 6= G, i.e. , H is a proper subgroup of G.
Example 2.18. For n ∈ Z+, An is a subgroup of Sn, i.e. , An 6 Sn.
By Example 2.3, for each choice of R = Q , R , C, there is a group (GL2( R) , ∗) with
½·
¸
¾
a b
GL2( R) =
: a, b, c, d ∈ R, ad − bc 6= 0 .
c d
Example 2.19. Let
½·
¸
¾
a b
SL2( R) =
: a, b, c, d ∈ R, ad − bc = 1 ⊆ GL
c d
2( R) .
Then SL2( R) is a subgroup of GL2( R), i.e. , SL2( R) 6 GL2( R).
Solution. This follows easily with aid of the three identities
·
¸
·
¸ − 1
·
¸
a b
a b
1
d
−b
det
= ad − bc;
det( AB) = det A det B;
=
.
¤
c d
c d
ad − bc −c
a
Let ( G, ∗) be a group. From now on, if x, y ∈ G we will write xy for x ∗ y. Also, for n ∈ Z
we write



 x( xn− 1)
if n > 0 ,
xn =
ι
if n = 0 ,


( x− 1) −n if n < 0 .
If g ∈ G,
hgi = {gn : n ∈ Z } ⊆ G
is a subgroup of G called the subgroup generated by g. This follows from the three equations
gmgn = gm+ n;
ι = g 0;
( gn) − 1 = g−n.
If hgi is finite and contains exactly n elements then g is said to have finite order |g| = n. If hgi is infinite then g is said to have infinite order |g| = ∞.
36
2. GROUPS AND GROUP ACTIONS Proposition 2.20. If g ∈ G has finite order |g| then
|g| = min {m ∈ Z+ : m > 0 , gm = ι}.
Example 2.21. In the group Sn the cyclic permutation ( i 1 i 2 · · · ir) of length r has order
|( i 1 i 2 · · · ir) | = r.
Solution. Setting σ = ( i 1 i 2 · · · ir), we have
( i
σk(1) =
k+1
if k < r,
i 1
if k = r,
hence |σ| 6 r. As ik 6= 1 for 1 < k 6 r, r is the smallest such power which is ι, hence |σ| = r. ¤
So for example, |(1 2) | = 2, |(1 2 3) | = 3 and |(1 2 3 4) | = 4. But notice that the product (1 2)(3 4 5) satisfies
((1 2)(3 4 5))2 = (1 2)(3 4 5)(1 2)(3 4 5) = (3 5 4) ,
hence |(1 2)(3 4 5) | = 6. On the other hand, the product (1 2)(2 3 4) satisfies
( (1 2)(2 3 4) )2 = (1 2)(2 3 4)(1 2)(2 3 4) = (1 3)(2 4)
so |(1 2)(3 4 5) | = |(1 3)(2 4) | = 2.
A group ( G, ∗) is called cyclic if there is an element c ∈ G such that G = hci; such a c is called a generator of G. Notice that for such a group, |G| = |c|.
Example 2.22. The group (Z , +) is cyclic of infinite order with generators ± 1.
Example 2.23. If 0 < n ∈ N0, then the group (Z /n, +) is cyclic of finite order n. Two
generators are ± 1 n ∈ Z /n. More generally, tn is a generator if and only if gcd( t, n) = 1.