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| Let S be a set and R be an equivalence relation on S. If a is an element of S, the equivalence class of a under R is the set {b in S|aRb} |
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| Let f be a map from A to B. f is one-to-one if for all a1, a2 in A, f(a1) =/= f(a2) |
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| Let g be a map from A to B. f is onto if for all b in B, there is some a in A such that g(a)=b |
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| A map that is one-to-one (injective) and onto (surjective) |
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| A binary operation on a set G is a map from GxG to G |
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| Let G have binary operation +. We say that + is associative is for g,h,k in G, g+(h+k)=(g+h)+k. |
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| Let G be a set with a binary operation +. Then e is an identity of G if e is in G and for all g in G, e+g=g+e=g |
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| Let G be a set with a binary operation +. Let e be the identity of G and let g be an element of G. Then an element h in G is an inverse of g if g+h=h+g=e. |
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| A group is a set that is associative, has an identity, and every element has an inverse. |
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Let G be a group and H be a subset of G. H is a subgroup of G if:
1) For all h1,h2 in H, h1h2 is in H
2) The identity of G is in H
3) For all h in H, the inverse of h is in H. |
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| Let G be a group. The center of G is the set Z(G)={z in G| for all G, gz=zg} |
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A group G is cyclic if there is some a in G such that G=
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Any a in G such that a G=
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| The number of elements in a group. |
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| The number of elements in the subgroup generated by the element. |
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| Given a set S, the symmetric group Sym(S) is the set of all bijections from S to S together with the operation of composition. |
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| A permutation of S is any element of Sym(S) |
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| Symmetric Group in n Letters |
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| A permutation (a1,a2,...,am) such that a1->a2, a2->a3, ..., am->a1, and it fixes all other elements of {1,2,...,n} |
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| Two cycles that don't interfere. |
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| The least common multiple of the lengths of the disjoint cycles |
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| Let a be in Sn. If a is a product of an even number of transpositions, it is an even permutation. Otherwise it is an odd permutation. |
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| The set of all even permutations of Sn, denoted An |
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| Let (G,*) and (H,+) be groups. An isomorphism is a map from G to H which is a bijection and is such that for all g1,g2 in G, we have f(g1*g2)=f(g1)+f(g2) |
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| Let G be a group. Then an automorphism of G is an isomorphism from G to G. |
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| Let G be a group and H be a subgroup of G. If g is in G, we denote gH={gh|h is in H}. gH is the left coset of g under H. A left coset of H in G is any set which is the left coset of g under H for some g in G. |
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| Let G be a finite group and let H be a subgroup of G. Then |H| divides |G|. |
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| Let p be a prime and let a be in Z. Then a^p=a mod p. |
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| Let G and H be groups. Then the external direct product of G an H is the set GxH = {(g,h)| g in G, h in H} with the operation (g1,h1)(g2,h2)=(g1g2, h1h2) for all g1,g2 in G and h1,h2 in H. |
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| Let G and H be groups. A group homomorphism from G to H is a map f:G->H such that for all g1,g2 in G we have f(g1,g2)=f(g1)f(g2). |
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| Let G, H be groups and f:G->H be a group homomorphism. Then the kernel of f is the set {g in G | f(g)=e} |
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| A subgroup H of g is normal if for all g in G, we have gH=Hg. |
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| Quotient Group (Factor Group) |
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| Let G be a group and let H be a normal subgroup of G. Then the quotient group G/H is the set of all left cosets of H in G together with the operation for all g1,g2 in G, (g1H)(g2H)=(g1g2)H |
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| First Isomorphism Theorem for Groups |
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Suppose G and H are groups and f:G->H is a homomorphism. Then:
1) Ker(f) is a normal subgroup of G
2) f(G) is a subgroup of H
3) The group G/ker(f) is isomorphic to f(G) |
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| A ring is a set R together with two binary operations + and * on R such that (R,+) is abelian and (R,*) is associative and for all s,r1,r2 in R we have s(r1+r2)=sr1+sr2 and (r1+r2)s=r1s+r2s. |
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| The unity of R is an element u in R such that ur=r=ru for all r in R. |
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| Let R be a ring with unity u. Then r in R is a unit of R if r has an inverse. |
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| A nonempty subset S of ring R is a subring if a-b and ab are in S whenever a and b are in S. |
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| Let R be a commutative ring. A nonzero element r in R is a zero divisor if there exists some s in R such that s is nonzero and rs=0. |
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| A commutative ring R with identity u such that u=/=0 and R does not have any zero divisors. |
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| An integral domain in which every nonzero element is a unit. |
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| The additive order of the identity. If it is infinite, characteristic is 0. |
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| A map f:R->S such that for all a,b in R we have f(a+b)=f(a)+f(b) and f(ab)=f(a)f(b). |
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A subset A of R such that
1) a-b is in A whenever a,b is in A
2) ra and ar are in A whenever a is in A and r is in R |
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| For ring R with ideal I, the quotient ring R/I is the set of additive left cosets of I in R, together with the operations (r1+I)+(r2+I)=(r1+r2)+I and (r1+I)(r2+I)=r1r2+I for all r1,r2 in R. |
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| First Isomorphism Theorem for Rings |
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| Let R and S be rings and let f:R->S be a ring homomorphism. Let k=ker(f). Then f(R) is a subring of S, ker(f) is an ideal of R, and f(R) is isomorphic to R/ker(f). |
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| Let a1,a2,...,an be in R. Then = {r1a1+r2a2+...+rnan | r1,r2,...rn is in R} is the ideal generated by a1,a2,...,an. |
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An ideal I of R is principal if there exists some a in I such that I =
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| An ideal P of R such that P=/=R and whenever a,b is in R and ab is in P, then either a is in P or b is in P. |
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| Let R be a commutative ring with identity. An ideal M of R is maximal if M=/=R if whenever I is an ideal of R and I contains M then either I=M or I=R. |
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