Group Theory

Question 1

General Linear Group

Answer 1

The set of all n*n invertible matrices with determinant not equal to 0 is a group under multiplication . Denoted as GL(n,F) where
G-General , L- linear , n- Order of Matrices , F - Set of entries in matrix
NOTE- Not Abelian as matrices are not abelian in general

Question 2

Special Linear Group

Answer 2

The set of all invertible n*n matrices having determinant =1 is a group under multiplication . Denoted as SL(n,F)
NOTE- Not Abelian as matrices are not abelian in general

Question 3

Klein 4 Group

Answer 3

(G,) is a group G={a,b,c,e} wrt * with following properties
ab=c , bc=a , ac=b & a^(-1)=a , b^(-1)=b , c^(-1)=c
denoted as K4/V4

Question 4

Unitory Group

Answer 4

The Group U(n) is defined as the set of all positive integers which are less than n, and are relatively prime with n . Under the binary operation “ multiplication modulo n”
eg : U(5)={1,2,3,4} with multiplication modulo 5

Question 5

Relationship between GL(n,F) and SL(n, F)

Answer 5

SL(n,F) is a proper Subset of GL(n,F)

Question 6

Dihedral Group (D_n /D_2n)

Answer 6

The set of all symmetries of a Polygon under the operation of composition , where D_n is no. of sides & D_2n is no. of elements in the group

Question 7

Properties of Dihedral Groups

Answer 7

1) r^n=1
2) s^2=1
3)s!=r^i for any i
4) rs=sr^(-1)
5)r^(i)s=sr^(-i)
6) Dihedral groups are non - abelian in general

Question 8

Group of n^th roots of unity

Answer 8

The set of n-roots of unity forms a group under multiplication

Question 9

Properties of Groups

Answer 9

1) identity element is unique
2)cancellation law hold in a group
3) the inverse of every element in a group is unique
4) In a group G , (ab)^-1=b^(-1)a^(-1) [Sock shoe property]

Question 10

Quaternion Group

Answer 10

(G,) is a group having 8 elements of the form
G= { 1 , -1 , i , -i , j ,-j , k ,-k }
and has following properties :
1)i^2=j^2=k^2=-1
2)ij=k , ji=-k
ki=j , ik =-j
jk=i , kj=-i
3)ij*k=1

Question 11

properties on order of an element

Answer 11

1). |a|=|a^-1|=m
2) if |a| is infinite then |a^(-1)| is also infinite .
3) (ab)^n= b^(n)a^(n)
=> (ab)^n!= a^(n).b^(n) in general & only holds true for abelian group
4) If |a|=m , |b|=n does not implies |ab|=mn( this only holds when GCD(m,n)=1 , then |ab|=mn

Question 12

One step subgroup test

Answer 12

Let H be a non- empty subset of a group G . Let a,b belongs to H then if a*b^(-1) also belongs to H ,
THEN H IS A SUBGROUP OF G

Question 13

Two Step Subgroup Test

Answer 13

Let H be a non- empty subset of a group G . Let a,b belongs to H then
If
1)a*b belongs to H &
2) a^-1belongs to H
THEN H IS A SUBGROUP OF G

Question 14

Finite Subgroup Test

Answer 14

Let H be a non empty finite subset of a group (G,*). Then if H has a closure under the operation of G then H is a subgroup of G.
a,b belongs to H
then if ab belongs to H for all a,b belonging to H

Question 15

Three important Subgroups

Answer 15

Centre , Centralizer , Normalizer

Question 16

Centre of a group

Answer 16

Let G be a Group (G,) and {g_1,g_2 ,g_3 …….. , g_n} be a set of elements which commute with every element of G . That is
g_1a=ag_1
g_2a=ag_2
.
.
.
g_na=ag_n
then the set of{ g_1,g_2,g_3,…..,g_n} is called as centre of (G,)
Denoted by Z(G)

Question 17

Is center if (G,*) subgroup of G

Answer 17

yes

Question 18

For any abelian group G , Z(G) is ?

Answer 18

Z(G)=G itself

Question 19

Centralizer of an element

Answer 19

For any element a belonging to G the centralizer of a , denoted by C(a) , is the set of all elements which commute with a.

Question 20

Can C(a) be empty ?

Answer 20

No , since identity is trivial

Question 21

Basic results on Centralizer of an element

Answer 21

1) Every element in Z(G) is always contained in C(a).
2) But those in C (a) need not be in Z(G)
3) intersection of all centralizers of elements in G = Z(G)

Question 22

What is a cyclic group?

Answer 22

A group G is said to be cyclic if every element of G has the form a^n , where a belongs to G and n belongs to Z .
a= Generator of the group G.
G= {a^n/n belongs to z}

Question 23

Is a group (z,+) cyclic?

Answer 23

Yes, 1 is the generator of the group

Question 24

Z_n is cyclic and 1 is generator of Z_n then all those elements , say m belongs to Z_n is also generator of Z_n if

Answer 24

GCD(m,n) =1

Question 25

If a generates Z_n , will a^(-1) also generate Z_n?

Answer 25

Yes Caution - this result doesn't implies that a group g has atleast 2 distinct generators a and a^(-1) Since a can be equal to a^(-1) .

Question 26

Let G be a finite cyclic group of order n , generated by a =G then |a|=?

Answer 26

n, since a is the generator so its order should be equal to order of the group .

Question 27

In a cyclic group If =G , then a becomes the generator of G and if not then a is ?

Answer 27

cyclic subgroup of G , since in a cyclic group G every element a belonging to G generates a cyclic group by itself , so if it is not a generator then it is a cyclic subgroup of G

Question 28

If G is not cyclic , then it doesn't have any cyclic subgroup TRUE/FALSE

Answer 28

False, even if G is not cyclic , G has a cyclic subgroup

Question 29

Every cyclic group is an abelian group TRUE/FALSE

Answer 29

True

Question 30

If G itself is a Cyclic group then every subgroup of G need not be cyclic TRUE / FALSE

Answer 30

False If G itself is a Cyclic group then every subgroup of G must be cyclic.

Question 31

what can you say about the nature of any finite cyclic group of order n & Z_n

Answer 31

Both have exactly same nature

Question 32

what can you say about the nature of any infinite cyclic group & Z

Answer 32

Both have exactly same nature

Question 33

In a cyclic group whose order is infinite , what is the condition for a^i=a^j

Answer 33

i=j

Question 34

In a cyclic group whose order is finite , what is the condition for a^i=a^j

Answer 34

if |a|= n , then a^i=a^j iff n divides (i-j)

Question 35

If |a|=n in a cyclic group , then a^k=e if ____

Answer 35

n divides k

Question 36

If a,b belongs to G , G being finite order cyclic group & ab=ba , then |ab| divides|a||b| TRUE/FALSE

Answer 36

True

Question 37

When does the cyclic group of order n generated by the elements a^i& a^j becomes equal ?

Answer 37

GCD(n,i)=GCD(n^j)

Question 38

Let |a|=n , then for any i belonging to z+ , the order of the element a^i belonging to is. ((((((((( technical issue Ignore this part

Answer 38

|a^i|= n/GCD(n,i)

Question 39

Fundamental theorem of cyclic groups

Answer 39

1) every subgroup of cyclic group is cyclic. 2)Let the order of cyclic group generated by an element a is n , then the order of any cyclic subgroup of is a divisor of n eg : |a|=48 then all other cyclic subgroups will have order 1,2,3,4,6,8,12,16,24,48.

Question 40

How many distinct subgroups of a given order exists ?

Answer 40

even if there exists distinct generators of a subgroup of |n| they all belong to only 1 unique subgroup

Question 41

The unique subgroup of order k of is generated by ____

Answer 41

a^(n/k) where ||=n

Question 42

Can we say in a cyclic group , |a|=n all a^i is also a generator of if i belongs to U(n) TRUE/ FALSE

Answer 42

True , since all elements in u(n) are relatively prime to n and will have GCD 1 with n.

Question 43

Euler ɸ Function

Answer 43

If for , |a|=n 1) if n is a prime no. then total number of generators of is ɸ (n)=n-1 2) if n is not prime then n=p1^m1p2^m2p3^m3.....pk^mk (factors of n ) ɸ(n)=ɸ(p1^m1)ɸ(p2^m2)ɸ(p3^m3).............ɸ(pk^mk) ɸ(p^m)=p^m-p^(m-1)

Question 44

The no. of elements of the order d in the cyclic group is ____

Answer 44

ɸ(d)

Question 45

In a finite group , the no. of elements of order d is ____

Answer 45

A multiple of ɸ(d)

Question 46

What is a symmetric group ?

Answer 46

Let A={1,2,3,4,....,n} Then , the set of all permutations of the elements of A is a group under composition is called as symmetric group . Represented as S_n where n= no. of elements .

Question 47

|S_n|=?

Answer 47

n!

Question 48

S_n is always non abelian & non cyclic when _____

Answer 48

n>= 3

Question 49

In cyclic notation of S_n we can change the order of appearance . TRUE/FALSE

Answer 49

FALSE

Question 50

In cyclic notation of S_n we can start with different entries . TRUE/FALSE

Answer 50

TRUE

Question 51

In cyclic notation of S_n we can start with different entries . TRUE/FALSE

Answer 51

TRUE

Question 52

Properties of permutation groups

Answer 52

1) Every permutation of S_n. can either be represented as a cycle OR a product of disjoint cycles. 2) Disjoint cycles always commute. 3)The order of an element in the S_n where α is written in the form of disjoint cycles / single cycle is |α|= LCM of length of its disjoint cycles.

Question 53

Let α =(123)(145) in S_5 Then , α^99 is a) (14532) b)(15423) c)(13254) d)(12534)

Answer 53

C

Question 54

Let α = (13597)(246)(810) If it is known that α^m is a 5 cycle , then which of the following values of m are possible? a)4 b)6 c)12 d)30

Answer 54

b,c

Question 55

Let α belongs to S_7 , where α^4= (2143567) then which of the following can be |α|? a)2 b)7 c)14 d)28

Answer 55

b important : why not 14 and 28 cause least no. of elements required for order 14 is 9 and for 28 is 11

Question 56

e is an _______ permutation (even / odd)

Answer 56

Even

Question 57

Results related to even and odd permutation

Answer 57

1) (Even)(Even)=(Even) 2) (Even)(odd)=(odd) 3) (odd)(odd)=(even) 4) If α is a cycle of length n then if n is odd=> α is even permutation n is even => α is odd permutation

Question 58

|A_n|=?

Answer 58

n!/2

Question 59

Claim 1) Let H = {Set of all even permutations in S_n } 2) Let K = {Set of all odd permutations in S_n } Then which of the following is/ are True a) H is a subgroup of S_n b) K is a subgroup of S_n c) H is not a subgroup of S_n d) K is not a subgroup of S_n

Answer 59

a,d because e is an even permutation and doesn't belong to K.

Question 60

What are the two necessary requirements for isomorphism?

Answer 60

1) Bijection 2) Operation Preserving

Question 61

What is isomorphism?

Answer 61

Let (G, *) and (G', .) be two groups Then G & G' are said to be ismomorphic , represented as G ≅G' If there exist a mapping ɸ: G -> G' such that 1)ɸ is one - one 2)ɸ is onto 3)ɸ is " operation preserving ". That is ɸ(a*b)=ɸa.ɸ(b)

Question 62

Let ɸ : (G,*) -> (G',.) be a isomorphism then , ɸ carries identity of G to _______ of G'.

Answer 62

Identity

Question 63

Let ɸ : (G,*) -> (G',.) be a isomorphism then , for any a belonging to G and n belonging to z ɸ(a^n)= __________

Answer 63

(ɸ(a))^n

Question 64

Let ɸ : (G,*) -> (G',.) be a isomorphism then, a&b commute in G iff ɸ (a) & ɸ (b) commute in G' True / False

Answer 64

True

Question 65

Let ɸ : (G,*) -> (G',.) be a isomorphism then, The group G is generated by an element a belonging to G implies G' is generated by _______

Answer 65

ɸ(a)

Question 66

Isomorphism do not preserve order of the element TRUE / FALSE

Answer 66

FALSE

Question 67

If G ≅ G' , G being finite ,then G & G' have exactly same number of element of each order True \ False

Answer 67

True

Question 68

G ≅ G' Then ,ɸ ^(-1) : G' -> G is not an ismorphism TRUE / FALSE

Answer 68

FALSE

Question 69

G ≅ G' then , if G is abelian -> G' is __________ ( abelian / non -abelian)

Answer 69

Abelian

Question 70

G ≅ G' then , if G is cyclic -> G' is __________ ( cyclic / non -cyclic)

Answer 70

Cyclic

Question 71

G ≅ G' , If H is a subgroup of G then ,ɸ (H) is not a subgroup of G' True / False

Answer 71

False, even vice versa for this statement is true

Question 72

G ≅ G' If z(G) is the centre of G , then ________ is the centre of G '

Answer 72

ɸ(z(G))

Question 73

G ≅ G' For any k belonging to z , b belonging to G the no. of solutions of 1) x^k=b 2) x^k=ɸ(b) are the same TRUE/ FALSE

Answer 73

TRUE

Question 74

WOTF are true ? 1) D_(2n) ≅ S_n, n>3 2)Z_(12) ≅ A_4 3) Z_(12 ) ≅ D_(12) 4) A_4 ≅ D _(12)

Answer 74

3,4 a is false cause different order after 3 b is false cause one is cyclic , other is not

Question 75

Cayley's Theorem

Answer 75

Every Group is isomorphic to a group of permutation that is Let g_i belongs to G for Tg_i = g_i *G , g belongs to G = {g_i* g_1 , g_i * g_2,......., g_i *g_n} where Tg_i is nothing but a permutation Then , the set of all such Tg_i i.e {Tg_1, T g_2, ......, Tg_n } is a group under composition which is isomorphic to G .

Question 76

What is automorphism ?

Answer 76

T : v. -> ɸ(G,*) -> (G,*) An automorphism form G onto G is an isomorphism which is defined from G onto G itself .

Question 76

What is automorphism ?

Answer 76

T:v->

Question 77

What is Inner Automorphism

Answer 77

Let G be a group , (G ,*) the function ɸ_a defined as, ɸ_a (x) = axa^(-1) is called as an inner automorphism induced by a belonging to G . It itself is a group under composition it is denoted by Inn(G)

Question 78

In the collection of all inner automorphisms , do all need to be distinct ?

Answer 78

No , in general collection of all inner automorphisms of the group need not be distinct .

Question 79

What is outer automorphisms?

Answer 79

Isomorphisms which are not inner .

Question 80

|Aut (Z_(10))|

Answer 80

4, as aut z(10)= {⌀ _(1->1) , ⌀_( 1->3) , ⌀_ (1->7), ⌀ _(1->9)} Aut (Z_n)≅U(n)

Question 81

Aut (S_n) ≅ S_n for n >=3 True /False

Answer 81

False, as it is not true for n=6 => exception is n=6

Question 82

Inn(S_n)≅ S_n True / False

Answer 82

True

Question 83

All Automorphism of S_n are not inner automorphism True / False

Answer 83

False, All Automorphism of S_n are inner automorphism

Question 84

S_6 has outer automorphisms where out (S_6) ≅ Aut (S_6) True / False

Answer 84

True

Question 85

|Aut(S_n) | , n!=6

Answer 85

n!

Question 86

|Aut S_6 |

Answer 86

2.6!= 1440

Question 87

|Aut D_n |

Answer 87

n ɸ(n) , where ɸ is Euler ɸ function

Question 88

The group of inner automorphism of D_n is

Answer 88

1) Isomorphic to D_n , when n is odd , i.e Inn(D_n)≅ D_n 2) Isomorphic to D_n/Z_2 , when n is even , i.e Inn (D_n ) ≅ D_n/ Z_2

Question 89

What is a coset?

Answer 89

Let (G,*) be a group , Let H be a subgroup of G , H =(g1,g2,g3....gn) Then the set of the form 1) aH=a*H is called as the left coset of H in G . 2) Ha= H*a is called as the right coset of H in G.

Question 90

What is a coset representative?

Answer 90

a belonging to aH or Ha , here a is called coset representative NOTE - Choice of coset representative i.e a in aH or Ha is not unique

Question 91

aH =H iff ______

Answer 91

a belongs to H => aH is a subgroup iff a belongs to H

Question 92

aH = bH iff _____

Answer 92

a belongs to bH / b belongs to aH OR ab^(-1) belongs to H

Question 93

is it true that either aH = bH or aH intersection bH = ⌀

Answer 93

True , either 2 cosets are exactly same or are completely disjoint

Question 94

aH = Ha iff H = aHa^(-1) True / False

Answer 94

True

Question 95

Langrange's Theorem

Answer 95

Let G be a finite group . Let H be a subgroup of G . Then 1) |H | divides |G| ( for finite group ) 2) The no. of Distinct left cosets / right cosets of H in G is |G|/|H| CAUTION - Converse of Langrange's Theorem is not true in General .

Question 96

Consequences of Langrange's Theorem

Answer 96

1) For any a belonging to G , G being finite , |a| divides |G| since ||=|a| & is a subgroup therefore |G| /|a| 2) If G has prime order then G is cyclic (converse fails) . 3) Let G be a finite group then a^|G|=e as a^|G|=a^{k|a|} =a^(mk)=(a^m)^k=e^k=e

Question 97

What is Index of a subgroup H in G?

Answer 97

It is the number of distinct left \ right costs of H in G , It is denoted as |G:H|=|G|/|H|

Question 98

Fermat's Little Theorem

Answer 98

For every integer a and every prime p , a^(p)mod p = a mod p ex: 3^7 mod 7 = 3 mod 7= 3

Question 99

For two finite subgroups H& K of Group G , where HK = {h*k / for all h belonging to H and k belonging to K} |HK|=_______

Answer 99

|H||K|/|H∩K| Caution : HK is a subgroup of G iff HK=KH

Question 100

Results for Automorphic group of A_n

Answer 100

1) For n != 2,3,6 Aut (A_n)≅S_n 2) Aut (A_3) ≅ Z_2 3) Aut (A_6)≅ Aut (S_6)

Question 101

Let G be a group of order 2p , p being prime>2 Then G ≅_____

Answer 101

Z_(2p) OR D_p(Dn)

Question 102

All subgroups of (R,+) are cyclic . True / False

Answer 102

False , as (Q,+) is a non -cyclic subgroup of (R,+)

Question 103

If f is a field then , every finite subgroup pf F={F-{0}, *} need not be cyclic . True / False

Answer 103

False

Question 104

The only subgroups of R* ={R-{0},*} having finite index |G|/|H| are _____&______

Answer 104

R* & R+

Question 105

No. of proper subgroups of R* having finite index is exactly ____

Answer 105

1, that is R+ Note 1) Index of R* in R* is 1 2) Index of R+ in R* is 2

Question 106

The group C* has no proper subgroup of finite index True / False

Answer 106

True , Only subgroup of C* having finite index is C* itself

Question 107

The group (Q,+) have no proper subgroup of finite index . True / False

Answer 107

True ,Only subgroup of(Q,+)having finite index is (Q,+) itself

Question 108

(R,+) have proper subgroup of finite order True / False

Answer 108

False

Question 109

All subgroups of (Q,+ ) are not cyclic True / False

Answer 109

True