Exam 1 Flashcards
Function
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.
Injective
A function f is injective if the following implication is true:
-if s, s’ in S satisfy f(s)=f(s’) then s=s’
Surjective
The function f is surjective if :
-for all t in T, there is an s in S with f(s)=t
Bijective
The function f is both injective and surjective
Associative operation
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)
Commutative operation
An operation * on S is a function :SxS —>S.
This operation is commutative if for all s1,s2 in S
s1s2=s2*s1
Distributive
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)
Identity element for (S,*)
The element e in S is an identity element for * if for all s in S:
es=s=se
Inverse element for (S,*) with identity e
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
Operation preserving mapping f:S —->T where (S, *) and (T, @) are sets equipped with operations
The mapping f:S —>T is operation preserving if for all a,b in S:
f(a*b)=f(a)@f(b)
Group
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*
Commutative group
A group is commutative if its operation is commutative
Symmetric group
The symmetric group on a set S is the set of all bijections of S equipped with the operation of composition
n-th dihedral group Dn
The n-th dihedral group is the set of distance preserving mappings of a regular n-gon
n divides m (n,m integers)
n divides m if there is an integer k such that m=kn
(k=l) mod n
This means that l=k divisible by n.
k mod n
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
Addition modulo n
We define
k +n l = (k+l) mod n
gcd(k,n)
The unique positive integer that divides k and n and is divisible by any other integer that divides k and n
lcm(k,n)
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
Isomorphism
An isomorphism between groups is an operation preserving bijection
Kernel of an operation preserving mapping
The kernel of an operation preserving mapping f: G —> K is the set
{g in G : f(g)=e}
Subgroup (H subset of G)
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.
Subgroup of G generated by g
The subgroup of G generated by g is the intersection of all subgroups of G that contain g.
Cyclic (group)
The group G is cyclic if it is equal to a subgroup of G generated by some single element of G
Order of an element of a group
The order of an element of a group is the number of elements of the subgroup generated by that element
Center
The center of G is the subgroup
Z(G)={a in G : for all g in G, ag=ga}
Relation
A relation ~ on S is a subset of SxS
Equivalence relation
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
Equivalence class (~ is an equivalence relation on G)
The equivalence class of g in G is the set
[g]={h in G: g ~ h}Right coset of H by g
The right coset of H by g is the set
Hg={hg : h in H}
