Exam 1

Question 1

Function

Answer 1

A function f: S —> T is a subset of SxT with the property that
-for all s in S, there is exactly one t in T such that (s,t) in f.

Question 2

Injective

Answer 2

A function f is injective if the following implication is true:
-if s, s’ in S satisfy f(s)=f(s’) then s=s’

Question 3

Surjective

Answer 3

The function f is surjective if :

-for all t in T, there is an s in S with f(s)=t

Question 4

Bijective

Answer 4

The function f is both injective and surjective

Question 5

Associative operation

Answer 5

An operation * on S is a function :SxS —>S.
This operation is associative if for all s1,s2,s3 in S:
(s1s2)s3=s1(s2*s3)

Question 6

Commutative operation

Answer 6

An operation * on S is a function :SxS —>S.
This operation is commutative if for all s1,s2 in S
s1s2=s2*s1

Question 7

Distributive

Answer 7

The operation @ is distributive with respect to * if for all s1, s2, s3,
s1 @ (s2s3)=(s1@s2)(s1@s3) and
(s2s3)@s1=(s2@s1)(s3@s1)

Question 8

Identity element for (S,*)

Answer 8

The element e in S is an identity element for * if for all s in S:
es=s=se

Question 9

Inverse element for (S,*) with identity e

Answer 9

The element s in S has an inverse with respect to * if there is another element s’ in S, such that
ss’=e=s’s

Question 10

Operation preserving mapping f:S —->T where (S, *) and (T, @) are sets equipped with operations

Answer 10

The mapping f:S —>T is operation preserving if for all a,b in S:
f(a*b)=f(a)@f(b)

Question 11

Group

Answer 11

A group is a set G equipped with an associated operation * such that,

• there is an identity element for *
• every element of G has an inverse with respect to*

Question 12

Commutative group

Answer 12

A group is commutative if its operation is commutative

Question 13

Symmetric group

Answer 13

The symmetric group on a set S is the set of all bijections of S equipped with the operation of composition

Question 14

n-th dihedral group Dn

Answer 14

The n-th dihedral group is the set of distance preserving mappings of a regular n-gon

Question 15

n divides m (n,m integers)

Answer 15

n divides m if there is an integer k such that m=kn

Question 16

(k=l) mod n

Answer 16

This means that l=k divisible by n.

Question 17

k mod n

Answer 17

the unique number r in {0,…n-1} such that there is an integer q with
k=qn+r
in other words, k mod n is the remainder left when dividing n by k

Question 18

Addition modulo n

Answer 18

We define

k +n l = (k+l) mod n

Question 19

gcd(k,n)

Answer 19

The unique positive integer that divides k and n and is divisible by any other integer that divides k and n

Question 20

lcm(k,n)

Answer 20

The unique positive number that is divisible by both k and n and divides any other integer that is divisible by both k and m

Question 21

Isomorphism

Answer 21

An isomorphism between groups is an operation preserving bijection

Question 22

Kernel of an operation preserving mapping

Answer 22

The kernel of an operation preserving mapping f: G —> K is the set
{g in G : f(g)=e}

Question 23

Subgroup (H subset of G)

Answer 23

H is a subgroup of G if the operation on G restricts to an operation on H and that operation makes H into a group.

Question 24

Subgroup of G generated by g

Answer 24

The subgroup of G generated by g is the intersection of all subgroups of G that contain g.

Question 25

Cyclic (group)

Answer 25

The group G is cyclic if it is equal to a subgroup of G generated by some single element of G

Question 26

Order of an element of a group

Answer 26

The order of an element of a group is the number of elements of the subgroup generated by that element

Question 27

Center

Answer 27

The center of G is the subgroup

Z(G)={a in G : for all g in G, ag=ga}

Question 28

Relation

Answer 28

A relation ~ on S is a subset of SxS

Question 29

Equivalence relation

Answer 29

An equivlanece relation ~ on S is a subset of SxS such that for all a,b,c in S,

• a~a
• if a~b then b~a
• if a~b and b~c then a~c

Question 30

Equivalence class (~ is an equivalence relation on G)

Answer 30

The equivalence class of g in G is the set
[g]={h in G: g ~ h}

Question 31

Right coset of H by g

Answer 31

The right coset of H by g is the set

Hg={hg : h in H}