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Summary Group Theory
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An overview of the key concepts and theorems of group theory at Utrecht University. It is based on the book Groups and Symmetry, Armstrong but not a perfect summary of the book.
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1 Group Theory1.1 Basic Concepts
Groups : A group is a set G together with an operation on G which satisfies:
Associativity; (xy)z = x(yz) ∀x, y, z ∈ G
There exists a unique identity element e ∈ G such that xe = x = ex ∀x ∈ G.
Each element x of G has a unique inverse x−1 ∈ G such that x−1 x = e = xx−1 .
−1 −1
For x1 , x2 , . . . , xn ∈ G holds (x1 x2 . . . xn )−1 = x−1
n . . . x2 x1 . We will denote
m copies of x as xm and n copies of x−1 as x−n . For m, n ∈ Z, xm xn = xm+n
and (xm )n = xmn with x0 = e.
Abelian Groups : A group is abelian (commutative) if xy = yx ∀x, y ∈ G.
Dihedral Groups : A dihedral group is the group of symmetries of a regu-
lar polygon (flat plates of n ≥ 3 equal sides for n ∈ Z) denoted by Dn . Let
r be a rotation of 2π/n through the axis perpendicular to the plate and s a
rotation of π about an axis in the plane of the plate. Then the elements of Dn
are: e, r, r2 , . . . , rn−1 , s, rs, r2 s, . . . , rn−1 s with rn = e, s2 = e and sr = r−1 s.
The infinite dihedral group D∞ is the group G consisting of the set of func-
tions from the real line to itself that preserve distance and sends the integers
among themselves.
Order : The order of a finite group is the number of elements in the group,
denoted by |G|.
Subgroup : A subgroup of a group G is a subset of G which forms a group
under the operation of G. If H is a subgroup of G we write H < G. For an el-
ement x ∈ G, the subgroup generated by x, < x > is the set of all powers of x.
Theorem 5.1 : A subset H is a subgroup of G if and only if xy −1 ∈ H
whenever x, y ∈ H.
Theorem 5.2 : The intersection of two subgroups of a group is itself a
subgroup.
Cyclic Groups : If there is an element x ∈ G that generates all of G
then G is a cyclic group.
Theorem 5.3 : Every subgroup of a cyclic group is cyclic.
1
,Words : Let x1 , . . . , xk ∈ X, X < G and m1 , . . . , mk ∈ Z. A word in
mk
the elements of X is xm 1 m2
1 x2 . . . xk . The collection of all words is a sub-
group of G. This group is called the subgroup generated by X and if it fills
out all of G, X is called a set of generators for G.
Permutations : A permutation of X is a bijection from X to itself. The
collection of all permutations of X forms the group SX . The non-abelian
group of permutations of n positive integers is denoted Sn , the symmetric
group.
An element of Sn is called odd if it consists of an odd number of transposi-
tions, idem for even. The even permutations of Sn form the alternating group
An . The 3-cycles generate An .
Cyclic Permutation : A ”k-cycle” (a1 , a2 , . . . , ak ) sends a1 to a2 , a2 to
a3 etc. A 2-cycle is called a transposition. The transpositions of Sn generate
Sn as do (12), (13), . . . , (1n).
Isomorphisms : The notion of similarity in group theory is expressed
through isomorphisms. Two groups G and G0 are isomorphic if there ex-
ists a bijection φ from G to G0 such that φ(xy) = φ(x)φ(y), ∀x, y ∈ G. To
indicate that two groups are isomorphic we write: G ∼ = G0 and G and G0
have the same order. An isomorphism sends the identity of G to that of G0
and sends inverses to inverses.
If H < G, then φ(H) < G0 . Furthermore, an isomorphism preserves the order
of an element. Compositions of isomorphisms are also isomorphisms. Exam-
ples are; cube ∼= S4 ∼= octahedron, tetrahedron ∼
= A4 and dodecahedron ∼ =
∼
A5 = isocahedron.
Cayley’s Theorem : Let G be a group, then G is isomorphic to a sub-
group of SG . If G is finite, it is isomorphic to a subgroup of Sn .
Matrix Groups : The set of all invertible n × n matrices (so det(A) 6= 0)
with entries in R forms the General Linear Group GLn . The set of orthogonal
matrices (At A = In ) forms the Orthogonal Group On . The elements of On
with det(A) = 1 form the Special Orthogonal Group SOn . The elements of
O2 with determinant 1 represent a rotation, the others represent a reflection.
2
, 1.2 Groups and Order
Direct Product : The direct product G × H of two groups G and H is a
group consisting of the ordered pairs (g, h), g ∈ G and h ∈ H with multipli-
cation defined as: (g, h)(g 0 , h0 ) = (gg 0 , hh0 ). If G and H are abelian, then so
is G × H and |G × H| = |G||H|.
Theorem 10.1 : Zm × Zn is cyclic if and only if the highest common factor
of m and n is 1.
Theorem 10.2 : If H and K are subgroups of G for which HK = G,
H ∩ K = e and hk = kh, ∀h ∈ H, k ∈ K, then G ∼
= H × K. Note that
HJ = {hj | h ∈ H, j ∈ J}.
Lagrange’s Theorem : The order of a subgroup of a finite group is al-
ways a divisor of the order of the group. If p is a prime divisor of the order
of the group, then the group contains a subgroup of order p.
Corollary 11.2 : The order of every element of G is a divisor of |G|.
Corollary 11.3 : If |G| is prime, G is cyclic.
Corollary 11.4 : If x ∈ G, then x|G| = e.
Relatively prime : Two integers n, m are relatively prime if the high-
est common factor (divisor) is 1.
Euler’s Totient Function : Euler’s totient function φ(n) denotes the num-
ber of integers that are relatively prime to n.
Euler’s Theorem : If the highest common factor of x and n is 1, then
xφ(n) ∼
= 1 mod(n).
Fermat’s Little Theorem : If p is prime and if x is not a multiple of
p, then xp−1 ∼
= 1 mod(p).
Cauchy’s Theorem : If p is a prime divisor of |G|, then G contains an
element of order p.
Theorem 13.2 : A group of order 6 is either isomorphic to Z6 or D3 .
3
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