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UNIT-V
GROUPS
PART- A
1. If S denotes the set of positive integers  100 for x , y  S define
x  y  minx , y.
Verify whether ( S , ) is a monoid assuming that  is associative.
Solution :
100 is the identity element in (s,x).
since 100*x = min {x,100}=x since x  100 for all x  S .
Therefore (s,x) is a monoid.
2. If H is a sub-group of a group G ,among the right coset of H in
G ,Prove that there is only one subgroup H .
Solution :
Let Ha be a right coset of H and G where a  G . If Ha is a
subgroup of G, then
e  Ha where e is the identity element in G.
3. Give an example of subsemigroup.
Solution :
For the semi-group (N,+) , the set E of all even non negative
integers is a sub semi
group (E,+) of (N,+).
4. Define Normal subgroup of a group.
A subgroup H of a group G is called a normal sub group if for every
a  G, aH  Ha
5. Find all the cosets of the subgroup H  1 ,1 in G  1 ,1 , i , i
with the operation
multiplication.
Solution :
Let us find the right cosets of H in G.
H 1  1, 1  H
H  1  1,1  H
H  i   i, i and H  i   i, i  Hi
H 1  H  H 1  1, 1
and Hi  H i  i, i
are the two right cosets of H in G. Similarly we can find the left coset
of H in G.
6. A semigroup homomorphism preserves property of associativity.
Solution :
Let a  S be an idempotent element.
There fore a*a = a
 g a * a  g a 
g a g a   g a 
This shows that g  a  is an idempotent element in T.
There fore the property of idempotency is preserved under semi group
homomorphism.
7. A semigroup homomorphism preserves commutativity.
Proof :
Let a, b  S
Assume that a * b  b * a
g  a *b   g b * a 
g  a  g b   g b  g  a 
This means that the operation is commutative in T.
There fore the semi group homomorphism preserves commutativity.
8. Define abelian group and subgroup.
Definition : Abelian group
A group (G,*) is said to be abelian if a*b=b*a for all a,b  G .
Definition : Subgroup
Let (G,*) be a group and let H be a non-empty subset of G. Then
H is said to
be a sub group of G if H itself is a group with the operation *.
9. Define ring and give an example of a ring with zero-divisors.
Definition : An algebraic system  S , ,* is called a ring if the binary
operations + and *
on S satisfy the following 3 properties.
i) (S,+) is an abelian group.
ii) (S,*) is a semi group.
iii) The operation * is distributive over + that is for any a, b, c  S
a.  b  c   a.b  a.c and
 b  c  .a  b.a  c.a
Example : The ring  Z10 , 10 , 10  not an integral domain. Since
5 10 2  0, yet 5  0, 2  0 in Z10 .
10.Define Encoder and Decoder.
Encoder : A device which is used to improve the efficiency of the
communication
channel is called an Encoder. It transforms the incoming
messages in such a
way that the presence of noise on the transformed message
is detectable.
Decoder : A decoder is used to transform the encoded message into
their original that is
acceptable to the receiver.
11.Define Hamming distance.
The number of places the two strings differ is called “Hamming distance
” of the two strings. The Hamming distance of the string x and y is denoted
by H(x,y).
12.State Cayley’s theorem on Permutation groups.
Every finite group of order ‘n’ is isomorphic to a permutation group
of degree n.
13.Prove that the only idempotent element of a group is its identity
element.
Idempotent element : An element a  G is said to be idempotent with
respect
to * if a*a=a
Identity element : An element e of G is said to be identity element if
e * a  a * e  a, for every a  G
In a group the identity element e only satisfies the idempotent
condition e*e=e.
Hence the proof.
14.What do you call a homomorphism of a semi group into itself?
Solution :
A homomorphism of a semi group into itself is called a semi group
endomorphism.
15.Show that if every element in a group is its own inverse, then the
group must
be abelian.
Solution :
Let  G,* be a group.
Given : a 1  a, a  G
To prove : a*b=b*a (Abelian)
Let a, b  G
 a 1  a and b 1  b
 Since  G,* be a group 
a *b G
a *b
1
 a *b
b 1 * a 1  a * b
 Since if every element in a group is its own inverse
 Since  a * b 1  b 1 * a 1 


by 1 
Since  G,* is an abelian group.
b * a  a *b
16.Give an example of a Monoid which is not a group.
 Z ,*

1
is a monoid which is not a group.  Since a  G,  G 

a

17.Show that  Z5 , 5  is a cyclic group.
[1] [2] [3] [4]
 5 [0]
____________________________
[0] 0
1
2
3
4
[1]
1
2
3
4
0
[2] 2
3
4
0
1
[3] 3
4
0
1
2
[4] 4
0
1
2
3
18.Define Semi group and monoid. Give an example of a semi group
which is not a monoid
Definition : Semi group
Let S be a non empty set and be a binary operation on S. The
algebraic system
(S, ) is called a semigroup if the operation is associative. In
other words (S, )
is semi group if for any x,y,z  G , (x y) z = x (y z).
Definition : Monoid
An algebraic system (M, ) is a Monoid if
i) is binary
ii) is associative
iii) the set M has an identity with respect to the operation .
Example : (Z, ) is a semi group.
The inverse property does not exist.
There fore (Z, ) is not a monoid.
19.A semigroup homomorphism preserves property of idempotency.
Let f   M ,*   H ,   be semi group homomorphism.
x idempotent element in M.
Therefore x * x  x, f  x * x   f  x  f  x 
Then f  x   f  x  f  x 
Therefore idempotency is preserved.
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