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Mathematics > Group Theory > Flashcards

Group Theory Flashcards

1
Q

General Linear Group

A

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

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2
Q

Special Linear Group

A

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

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3
Q

Klein 4 Group

A

(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

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4
Q

Unitory Group

A

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

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5
Q

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

A

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

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6
Q

Dihedral Group (D_n /D_2n)

A

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

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7
Q

Properties of Dihedral Groups

A

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

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8
Q

Group of n^th roots of unity

A

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

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9
Q

Properties of Groups

A

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]

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10
Q

Quaternion Group

A

(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

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11
Q

properties on order of an element

A

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

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12
Q

One step subgroup test

A

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

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13
Q

Two Step Subgroup Test

A

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

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14
Q

Finite Subgroup Test

A

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

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15
Q

Three important Subgroups

A

Centre , Centralizer , Normalizer

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16
Q

Centre of a group

A

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)

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17
Q

Is center if (G,*) subgroup of G

A

yes

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18
Q

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

A

Z(G)=G itself

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19
Q

Centralizer of an element

A

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.

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20
Q

Can C(a) be empty ?

A

No , since identity is trivial

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21
Q

Basic results on Centralizer of an element

A

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)

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22
Q

What is a cyclic group?

A

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}

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23
Q

Is a group (z,+) cyclic?

A

Yes, 1 is the generator of the group

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24
Q

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

A

GCD(m,n) =1

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25
Q

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

A

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) .

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26
Q

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

A

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

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27
Q

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

A

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

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28
Q

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

A

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

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29
Q

Every cyclic group is an abelian group
TRUE/FALSE

A

True

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30
Q

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

A

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

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31
Q

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

A

Both have exactly same nature

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32
Q

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

A

Both have exactly same nature

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33
Q

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

A

i=j

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34
Q

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

A

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

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35
Q

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

A

n divides k

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36
Q

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

A

True

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37
Q

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

A

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

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38
Q

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

A

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

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39
Q

Fundamental theorem of cyclic groups

A

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 <a> 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.</a>

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40
Q

How many distinct subgroups of a given order exists ?

A

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

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41
Q

The unique subgroup of order k of<a> is generated by ____</a>

A

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

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42
Q

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

A

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

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43
Q

Euler ɸ Function

A

If for <a> , |a|=n
1) if n is a prime no. then total number of generators of <a> 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)</a></a>

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44
Q

The no. of elements of the order d in the cyclic group <a> is ____</a>

A

ɸ(d)

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45
Q

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

A

A multiple of ɸ(d)

46
Q

What is a symmetric group ?

A

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 .

47
Q

|S_n|=?

A

n!

48
Q

S_n is always non abelian & non cyclic when _____

A

n>= 3

49
Q

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

A

FALSE

50
Q

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

A

TRUE

51
Q

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

A

TRUE

52
Q

Properties of permutation groups

A

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.

53
Q

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

A

C

54
Q

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

A

b,c

55
Q

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

A

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

56
Q

e is an _______ permutation (even / odd)

A

Even

57
Q

Results related to even and odd permutation

A

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

58
Q

|A_n|=?

A

n!/2

59
Q

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

A

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

60
Q

What are the two necessary requirements for isomorphism?

A

1) Bijection
2) Operation Preserving

61
Q

What is isomorphism?

A

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(ab)=ɸa.ɸ(b)

62
Q

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

A

Identity

63
Q

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

A

(ɸ(a))^n

64
Q

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

A

True

65
Q

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 _______

A

ɸ(a)

66
Q

Isomorphism do not preserve order of the element
TRUE / FALSE

A

FALSE

67
Q

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

A

True

68
Q

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

A

FALSE

69
Q

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

A

Abelian

70
Q

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

A

Cyclic

71
Q

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

A

False, even vice versa for this statement is true

72
Q

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

A

ɸ(z(G))

73
Q

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

A

TRUE

74
Q

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)

A

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

75
Q

Cayley’s Theorem

A

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 .

76
Q

What is automorphism ?

A

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

76
Q

What is automorphism ?

A

T:v->

77
Q

What is Inner Automorphism

A

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)

78
Q

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

A

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

79
Q

What is outer automorphisms?

A

Isomorphisms which are not inner .

80
Q

|Aut (Z_(10))|

A

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

81
Q

Aut (S_n) ≅ S_n for n >=3
True /False

A

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

82
Q

Inn(S_n)≅ S_n
True / False

A

True

83
Q

All Automorphism of S_n are not inner automorphism
True / False

A

False, All Automorphism of S_n are inner automorphism

84
Q

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

A

True

85
Q

|Aut(S_n) | , n!=6

A

n!

86
Q

|Aut S_6 |

A

2.6!= 1440

87
Q

|Aut D_n |

A

n ɸ(n) , where ɸ is Euler ɸ function

88
Q

The group of inner automorphism of D_n is

A

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

89
Q

What is a coset?

A

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=aH is called as the left coset of H in G .
2) Ha= H*a is called as the right coset of H in G.

90
Q

What is a coset representative?

A

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

91
Q

aH =H iff ______

A

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

92
Q

aH = bH iff _____

A

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

93
Q

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

A

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

94
Q

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

A

True

95
Q

Langrange’s Theorem

A

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 .

96
Q

Consequences of Langrange’s Theorem

A

1) For any a belonging to G , G being finite , |a| divides |G|
since |<a>|=|a| & <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</a></a>

97
Q

What is Index of a subgroup H in G?

A

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

98
Q

Fermat’s Little Theorem

A

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

99
Q

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|=_______

A

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

100
Q

Results for Automorphic group of A_n

A

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

101
Q

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

A

Z_(2p) OR D_p(Dn)

102
Q

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

A

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

103
Q

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

A

False

104
Q

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

A

R* & R+

105
Q

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

A

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

106
Q

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

A

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

107
Q

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

A

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

108
Q

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

A

False

109
Q

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

A

True

Mathematics (6 decks)

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