1. GROUPS
ADVANCED GROUP THEORY
References
GROUP THEORY
Shinoj K.M.
Department of Mathematics
St.Josephās College,Devagiri
Kozhikode-8
shinusaraswathy@gmail.com
8 December 2018
Shinoj K.M. Department of MathematicsGROUP THEORY
2. GROUPS
ADVANCED GROUP THEORY
References
Outline
1 GROUPS
Order
Cyclic Groups
Normal Subgroups
2 ADVANCED GROUP THEORY
Direct product of groups
Fundamental Theorem of ļ¬nite Abelian groups
Classiļ¬cation of groups of order up to 10
Shinoj K.M. Department of MathematicsGROUP THEORY
3. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Deļ¬nition
A set G together with a binary operation ā is called a group if
G is closed w.r.t. ā
ā is associative
there exists an identity element in G
for each element in G there is an inverse in G
If ā is commutative, then G is an Abelian group.
If ā is not commutative, then G is called a non-Abelian group.
Shinoj K.M. Department of MathematicsGROUP THEORY
4. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Example
(Z,+) is an Abelian group.
(Zn, +n) is an Abelian group.
The set of all nth roots of unity, Un, is an Abelian group
w.r.t. multiplication.
The set of all n Ć n invertible matrices with real entries,
GL(n, R), called the general linear group, is a non-Abelian
group , w.r.t. matrix multiplication.
Shinoj K.M. Department of MathematicsGROUP THEORY
5. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
The symmetric group on n letters Sn , n ā„ 3 is a
non- Abelian group.
The group table of S3
* Ļ0 Ļ1 Ļ2 Āµ1 Āµ2 Āµ3
Ļ0 Ļ0 Ļ1 Ļ2 Āµ1 Āµ2 Āµ3
Ļ1 Ļ1 Ļ2 Ļ0 Āµ3 Āµ1 Āµ2
Ļ2 Ļ2 Ļ0 Ļ1 Āµ2 Āµ3 Āµ1
Āµ1 Āµ1 Āµ2 Āµ3 Ļ0 Ļ1 Ļ2
Āµ2 Āµ2 Āµ3 Āµ1 Ļ2 Ļ0 Ļ1
Āµ3 Āµ3 Āµ1 Āµ2 Ļ1 Ļ2 Ļ0
Shinoj K.M. Department of MathematicsGROUP THEORY
6. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Fact
The equation x2 = e has 4 solutions in Klein-4 group.
The equation x2 = ā1 has 6 solutions in Q8.
So a polynomial equation of degree n can have
more than n solutions in a group.
Shinoj K.M. Department of MathematicsGROUP THEORY
7. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Order
Deļ¬nition
The order of an element a in a group is the smallest
positive integer n such that an = e.
The order of a group is the number of elements in the
group.
Shinoj K.M. Department of MathematicsGROUP THEORY
8. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Fact
If O(G) is even, then there is atleast one element in G of
order 2.
In a ļ¬nite group, all elements are of ļ¬nite order.
There are inļ¬nite groups with all elements of ļ¬nite order.
Shinoj K.M. Department of MathematicsGROUP THEORY
9. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Deļ¬nition
Let G be an abelian group.
Then Gt = {u ā G/O(u) is ļ¬nite }, is a subgroup of G,
called the torsion subgroup of G.
If Gt = {0}, then G is said to be torsion free.
If Gt = G, then G is called a torsion group.
Shinoj K.M. Department of MathematicsGROUP THEORY
10. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Example
Zn and all ļ¬nite Abelian groups are torsion groups.
Z and Z Ć Z are torsion free groups.
Z Ć Z2 is neither torsion group nor torsion free group.
Shinoj K.M. Department of MathematicsGROUP THEORY
11. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
If G is a non-abelian group, there can be two
elements of ļ¬nite order whose product is of inļ¬nite
order.
Shinoj K.M. Department of MathematicsGROUP THEORY
12. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
If G is a non-abelian group, there can be two
elements of ļ¬nite order whose product is of inļ¬nite
order.
Example(i): Consider the following two elements in
GL(2, R).
A =
ā1 0
0 1
and B =
ā1 ā1
0 1
Example(ii): Consider the group G of all permutations on
R, the set of all real numbers and f(x) = āx and
g(x) = 1 ā x.
Shinoj K.M. Department of MathematicsGROUP THEORY
13. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Theorem (Cauchy)
Let G be a ļ¬nite group and let p be a prime number
dividing O(G). Then G has an element of order p and
consequently G has a subgroup of order p.
Shinoj K.M. Department of MathematicsGROUP THEORY
14. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Let O(G) = n and d divides n. Does there always exist
an element in G of order d ?
Shinoj K.M. Department of MathematicsGROUP THEORY
15. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Let O(G) = n and d divides n. Does there always exist
an element in G of order d ?
Need not!! Look at the following examples.
Klein-4 group.
S3
Z2 Ć Z4.
Z2 Ć Z2 Ć Z2
Shinoj K.M. Department of MathematicsGROUP THEORY
16. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Cyclic Groups
Deļ¬nition
An inļ¬nite group is cyclic iļ¬ it is isomorphic to (Z,+).
An inļ¬nite group is cyclic if and only if it is isomorphic to
all its nontrivial subgroups.
A ļ¬nite group G is cyclic iļ¬ G has an element of order d,
for each d dividing o(G).
Shinoj K.M. Department of MathematicsGROUP THEORY
17. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Fact
(Zn, +n) and (Z, +) prototypes of cyclic groups.
Every cyclic group is abelian.
If m is a square free integer, then every Abelian group of
order m is cyclic.
The smallest possible order of a non cyclic group is 4.
Shinoj K.M. Department of MathematicsGROUP THEORY
18. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Theorem
The number of subgroups of a cyclic group of order n is the
number of divisors of n.
The number of generators of (Zn, +n) is Ļ(n), the number
of positive integers less than and relatively prime to n.
A cyclic group with exactly one generator can have atmost
2 elements.
Every group of prime order is cyclic.
Shinoj K.M. Department of MathematicsGROUP THEORY
19. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Fact
Every group is the union of its cyclic subgroups.
A group is ļ¬nite if and only if it has only ļ¬nitely many
subgroups.
Shinoj K.M. Department of MathematicsGROUP THEORY
20. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Normal Subgroups
Deļ¬nition
A subgroup H is a normal subgroup of G if ghgā1 ā H,
ā g ā G and ā h ā H.
A subgroup H of a group G is normal iļ¬ aH = Ha,
ā a ā G.
Shinoj K.M. Department of MathematicsGROUP THEORY
21. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Example
(2Z, +) is a normal subgroup of (Z, +)
SL(n, R) is a normal subgroup of GL(n, R).
An is a normal subgroup of Sn.
Shinoj K.M. Department of MathematicsGROUP THEORY
23. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Theorem
Every subgroup of an abelian group is normal.
Question:Let G be a group such that all its
subgroups are normal. Can we conclude that G is
abelian?
Shinoj K.M. Department of MathematicsGROUP THEORY
24. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Theorem
Every subgroup of an abelian group is normal.
Question:Let G be a group such that all its
subgroups are normal. Can we conclude that G is
abelian?
Answer: NO
Counter example : Q8
Shinoj K.M. Department of MathematicsGROUP THEORY
25. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Theorem
Lagrangeās Theorem:
Let G be a ļ¬nite group and H be a subgroup of G. Then O(H)
divides O(G).
The converse of Lagrangeās Theorem is true for Abelian
groups.
Shinoj K.M. Department of MathematicsGROUP THEORY
26. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Question: Suppose G is a ļ¬nite group and it has
subgroups of order d for each d dividing n. Can we
conclude that G is abelian?
Shinoj K.M. Department of MathematicsGROUP THEORY
27. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Question: Suppose G is a ļ¬nite group and it has
subgroups of order d for each d dividing n. Can we
conclude that G is abelian?
Answer : NO.
Counter examples are
S3
The Quartenion group, Q8
Shinoj K.M. Department of MathematicsGROUP THEORY
28. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Theorem
The converse of Lagrangeās Theorem is not generally true.
A4 has no subgroup of order 6.
Proof.
Let H be a subgroup of order 6.
Let a ā A4, be of order 3.
Consider the cosets H, aH, a2H. Since o(H) = 6 atleast two of
these cosets must be equal.
H = aHā aāH
aH = a2H ā a2aā1 ā H ā a ā H
H = a2H ā a2 ā H ā a ā H
Thus we get that all elements of order 3,must be in H,which is
a contradiction.(Since there are 8 elements of order 3 in A4).
So A4 cannot have a subgroup of order 6.
Shinoj K.M. Department of MathematicsGROUP THEORY
29. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Theorem
If H is a normal subgroup of G Then the set
G/H = {aH; a ā G} is a group, called factor group, under
the operation (aH)(bH) = abH.
Shinoj K.M. Department of MathematicsGROUP THEORY
30. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Deļ¬nition
A homomorphism Ļ from a group G to ĀÆG is a mapping
from G into ĀÆG that preserves the group operation.
Ker Ļ = {x ā G/Ļ(x) = e} .
Shinoj K.M. Department of MathematicsGROUP THEORY
31. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Theorem
Ker Ļ is a normal subgroup of G.
G/KerĻ Ļ(G).
Every normal subgroup of a group G is the kernel of a
homomorphism of G.
Shinoj K.M. Department of MathematicsGROUP THEORY
33. GROUPS
ADVANCED GROUP THEORY
References
Order
Cyclic Groups
Normal Subgroups
Theorem
Let F be a ļ¬nite ļ¬eld of q elements. GL(n, q) is the group
of all n Ć n invertible matrices with entries from F.Then
|GL(n, q)| = (qn ā 1)(qn ā q)(qn ā q2). . .(qn ā qnā1).
SL(n, q) is the group of all n Ć n matrices having
determinant 1,with entries from F.Then
|SL(n, q)| =
(qn ā 1)(qn ā q)(qn ā q2). . .(qn ā qnā1)
q ā 1
.
Shinoj K.M. Department of MathematicsGROUP THEORY
34. GROUPS
ADVANCED GROUP THEORY
References
Direct product of groups
Fundamental Theorem of ļ¬nite Abelian g
Classiļ¬cation of groups of order up to 10
Direct product of Groups
Consider the group G1 Ć G2, where G1 and G2 are groups.
If H1, H2 are subgroups of G1 , G2 respectively then
H1 Ć H2 is a subgroup of G1 Ć G2.
Is every subgroup of G1 Ć G2 is of the form H1 Ć H2?
Shinoj K.M. Department of MathematicsGROUP THEORY
35. GROUPS
ADVANCED GROUP THEORY
References
Direct product of groups
Fundamental Theorem of ļ¬nite Abelian g
Classiļ¬cation of groups of order up to 10
Theorem
The group Zm Ć Zn is isomorphic to Zmn iļ¬ m and n are
relatively prime.
Shinoj K.M. Department of MathematicsGROUP THEORY
36. GROUPS
ADVANCED GROUP THEORY
References
Direct product of groups
Fundamental Theorem of ļ¬nite Abelian g
Classiļ¬cation of groups of order up to 10
Theorem
Fundamental Theorem of ļ¬nite abelian groups:
Every ļ¬nite abelian group is a direct product of cyclic groups of
prime-power order. Moreover, the factorisation is unique except
for rearrangement of factors.
Shinoj K.M. Department of MathematicsGROUP THEORY
37. GROUPS
ADVANCED GROUP THEORY
References
Direct product of groups
Fundamental Theorem of ļ¬nite Abelian g
Classiļ¬cation of groups of order up to 10
Theorem
If there are r Abelian groups of order m and s Abelian
groups of order n and g.c.d.(m, n) = 1, then there are rs
Abelian groups of order mn.
Let N = pn1
1 pn2
2 ...pnk
k , where piās are distinct primes
dividing N. Then the number of abelian groups of order N
is P(n1)P(n2)...P(nk) where P(ni) is the number of
partitions ni.
Shinoj K.M. Department of MathematicsGROUP THEORY
38. GROUPS
ADVANCED GROUP THEORY
References
Direct product of groups
Fundamental Theorem of ļ¬nite Abelian g
Classiļ¬cation of groups of order up to 10
How many abelian groups are there of order 4?
How many abelian groups are there of order 8?
How many abelian groups are there of order 100?
How many abelian groups are there of order 600?
How many abelian groups are there of order 1000?
How many abelian groups are there of order 105?
Shinoj K.M. Department of MathematicsGROUP THEORY
39. GROUPS
ADVANCED GROUP THEORY
References
Direct product of groups
Fundamental Theorem of ļ¬nite Abelian g
Classiļ¬cation of groups of order up to 10
Theorem
For a prime p, every group of order p2 is abelian.
For an odd prime p, every group of order 2p is either Z2p
or the dihedral group Dp.
Shinoj K.M. Department of MathematicsGROUP THEORY
40. GROUPS
ADVANCED GROUP THEORY
References
Direct product of groups
Fundamental Theorem of ļ¬nite Abelian g
Classiļ¬cation of groups of order up to 10
Theorem
There are exactly two distinct non-Abelian groups of order
8, the Quartenion group Q8 and the dihedral group D4.
Shinoj K.M. Department of MathematicsGROUP THEORY
41. GROUPS
ADVANCED GROUP THEORY
References
Direct product of groups
Fundamental Theorem of ļ¬nite Abelian g
Classiļ¬cation of groups of order up to 10
Order Abelian Groups Non-Abelian Groups
1 Z1
2 Z2
3 Z3
4 Z4, Z2 Ć Z2
5 Z5
6 Z6 D3
7 Z7
8 Z8, Z4 Ć Z2, Z2 Ć Z2 Ć Z2 D4, Q8
9 Z9, Z3 Ć Z3
10 Z10 D5
Shinoj K.M. Department of MathematicsGROUP THEORY
42. GROUPS
ADVANCED GROUP THEORY
References
REFERENCES
[1] Joseph A. Gallian ,āContemporary Abstract Algebraā ,
Narosa Publishing House.
[2] John.B.Fraleigh, āA First Course in Abstract Algebraā.
[3] Thomas V.Hungerford, āAbstract Algebraā
Shinoj K.M. Department of MathematicsGROUP THEORY