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Abstract Algebra Cheat Sheet

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This document provides definitions and propositions related to abstract algebra. It begins by defining a group as a set with a binary operation that is closed, associative, has an identity element, and where each element has an inverse. It then lists several propositions about properties of groups, including that a group has a unique identity and each element has a unique inverse. The document continues defining additional algebraic structures like rings, fields, subgroups, and properties of groups like cyclic groups. It concludes by discussing matrix groups and their properties.

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Abstract Algebra Cheat Sheet 16 December 2002 By Brendan Kidwell, based on Dr. Ward Heilman’s notes for his Abstract Algebra class. Notes: Where applicable, page numbers are listed in parentheses at the end of a note. Def: A group is a nonempty set G together with a binary operation 2 on G3G satisfying the following four properties: 1. G is closed under the operation 2 . 2. The operation 2 is associative. 3. G contains an identity element, e, for the operation 2 . 4. Each element in G has an inverse in G under the operation 2 . Proposition 1: A group has exactly one identity element. Proposition 2: Each element of a group has exactly one inverse element. Proposition 3: ‘a2b’ 1,b 12a 1 } a, bV‘G, 2’ .       Proposition 4: ‘a 1 ’ 1,a } aV‘G, 2’ .     Proposition 5: ‘én , 0 n ’ is a group } nVè . Proposition 6: In a group table, every element occurs exactly once in each row and exactly once in each column. Def: The order of a group ‘G, 2’ is the number of elements in the set G. (Written as bGb .) (36) Def: A dihedral group of order 2n is the set of symmetric transformations of a regular n -gon . (Written as D n .) (36) Def: An abelian (or commutative) group has the property that a2b,b2a } a, bV‘G, 2’ . (37) Def: ‘H, 2’ is a subgroup of ‘G, 2’ iff H VG and ‘H, 2’ is a group under the same operation. (37) To show that ‘H, 2’ is a subgroup, show that H VG and then show closure and existence of inverses. Lagrange’s Theorem: Let ‘H, 2’ be a subgroup of a finite group, ‘G, 2’ . bHb divides bGb . Def: “ a ”,a 0 , a1 , a 1 , a 2 , a 2 , a3 , a 3 e is the cyclic subgroup generated by a.       Def: The order of an element, a, is the order of “ a” . Def: A cyclic group is a group that can be generated entirely by repeatedly combining a single element with itself. In other words, if for a cyclic group G,“ a” , then a is the generator of G. Def: Prime Order Proposition. For every prime p, there is exactly one group of order p. Proposition 8: Cancellation Laws. Let a, b, cV‘G, 2’ . 1. ‘a2b,a2c’ l‘b,c’ 2. ‘b2a,c2a’ l‘b,c’ 3. If G is abelian, ‘a2b,c2a’ l‘b,c’ Proposition 9: The only solution to a2a,a is a,e . Proposition 10: Let a, b V G . If a2b/b2a , then e, a, b, a2b, b2a are all distinct elements. (50)
Proposition 11: Any non-abelian group has at least six elements. (51) Def: The center of a group is Z ‘G’,all g VG such that ‘ g2a,a2g } aVG ’ . Proposition 12: ‘Z ‘G’, 2’ is a subgroup of G. (52) Def: Two integers, a and b, are relatively prime iff gcd ‘a, b’,1 . (54) Def. } nVè , the set of units of n, U ‘n’ , is the set of all natural numbers relatively prime to n. (54) Proposition 13: } nVè , ‘U ‘n’,4n ’ is a group. (54) Def: For any set S and subsets A, B VS , the symmetric difference of A and B (written as A´ B ) is the set of all elements that are in A or B, but are not in both A and B. In other words, A ´ B ,‘ A1 B ’=‘ B 1 A ’ . (55) Def: The power set of S (written as P‘S ’ ) is the set of all subsets of S, including Ž and the original set S. (55) Proposition 14: For any nonempty set S, ‘ P‘S ’, ´ ’ is a group. (55) Def: Let ‘G, 2’ and ‘K, ( ’ be two groups. Let f be a function from G to K. f is a homomorphism (or operation preserving function) from ‘G, 2’ to ‘K, ( ’ iff } a, bVG f ‘a2b’, f ‘a’( f ‘b’ . (59) Proposition 15: Let f: G l K be a homomorphism. Let e be the identity of ‘G, 2’ and e ’ be the identity of ‘K, ( ’ . (60) 1. f ‘e’,e ’ 2. f ‘ g 1 ’,‘ f ‘ g ’’ 1 } g V G ¡ ¡ 3. f ‘g n ’,‘ f ‘g’’n } nV é Def: Given nonempty sets S and T, with x, y V S , and a function f: S lT (63) 1. f is a one-to-one (1-1) function iff ‘ x/ y ’ l ‘ f ‘x’/ f ‘ y’’ . 2. f is onto T iff } z VT YxV S such that f ‘x’, z . Proposition 16: Let f: S lT be an onto function. (65) 1. f ‘ f 1 ‘V ’ ’,V }V PT ¢ 2. W P f ‘ f 1 ‘W ’’ } W PS ¢ Proposition 17: Let f be a homomorphism from ‘G, 2’ to ‘K, ( ’ . (68) 1. If ‘H, 2’ is a subgroup of ‘G, 2’ , then ‘ f ‘H ’, ( ’ is a subgroup of ‘K, ( ’ . 2. If ‘L, ( ’ is a subgroup of ‘K, ( ’ , then ‘ f 1 ‘L’, 2 ’ is a subgroup of ‘G, 2’ . ¢ Def: (Using the previous example,) the image of H under f is f ‘H ’ . The inverse image of L under f is f 1 ‘L’ . (68) ¢ Proposition 18: Let f be a homomorphism from ‘G, 2’ to ‘K, ( ’ . f is one-to-one iff ker ‘ f ’,e  . (72) Def: Two groups, ‘G, 2’ and ‘K, ( ’ , are isomorphic iff there exists a one-to-one homomorphism f from ‘G, 2’ onto ‘K, ( ’ —that is, f ‘G’,K . In this case, f is called an isomorphism or isomorphic mapping. (73) Proposition 19: Every finite cyclic group of order n is isomorphic to ‘é n, 0 n ’ and every infinite cyclic group is isomorphic to ‘é , 0 ’ . (75) Proposition 20: Every subgroup of a cyclic group is cyclic. (76)
Theorem: If G is a finite group, p is a prime, and p k is the largest power of p which divides bGb , then G has a subgroup of order pk . Def: A permutation is a one-to-one and onto function from a set to itself. (77) Note: See pages 78 and 81 for examples of how to notate permutations. Def: The set of permutations on 1, 2, 3,e, n is written as S n . (79) Theorem 21: The set of all permutations together with composition, ‘S n, ( ’ , is a nonabelian group } nB3 . (79) Theorem 22: The set of all permutations on a set S (its symmetries), together with composition, ‘Sym S, ( ’ , is a group. (80) Theorem 23 (Cayley’s Theorem): Every group is isomorphic to a group of permutations. (82) Proposition 24: Every permutation can be written as a product of disjoint cycles in permutation notation. (86) Def: The length of a cycle in a permutation is the number of distinct objects in it. A cycle of length 2 is a transposition. (86) Proposition 25: Every cycle can be written as a product of transpositions (not necessarily distinct). (87) Def: A permutation is even (or odd) if it can be written as a product of an even (or odd) number of transpositions. (88) Def: The subset of S n which consists of all the even permutations of S n is called the alternating group on n and is written as An . (90) Def: Matrix multiplication, which is not commutative, is the standard way to combine matrices. To multiply a 2×2 matrix: (102) • –• – • a b e f c d g h , a e0b g a f 0b h c e0d g c f 0d h – Notes: A 2×2 matrix can be found to represent any linear transformation. The special matrix M, •cos ¾ 1sin ¾ sin ¾ cos ¾ – when mulpilied on the left with a YHFWRU LQ ë 2 ZLOO URWDWH LW FRXQWHUFORFNZLVH by the amount ¾ : M X initial , X rotated . (100) Def: The inverse under multiplication of a 2×2 matrix is computed as follows: (103) • – d 1b 1 • – a b a d1b c a d1b c £ , c d 1c a a d1b c a d1b c Def: The determinant of a 2×2 matrix is computed as follows: (104) det ‘• –’ a b c d ,a d1b c Def: A matrix is invertible iff its determinant is nonzero. (104) Theorem 29: The set of all invertible 2×2 made from elements of ë , together with matrix multiplication, forms a group, called the general linear group, which is written as GL‘2, ë’ . (105)
Def: The special linear group is the group of 2×2 matrices with determinants of 1, written as SL‘2, ë’ . (106) Def: To get the transpose of a matrix, swap each element a i, j with the one on the opposite side of the main diagonal, a j, i . The transpose of a matrix M is written M t . (106) Def: A matrix M is orthogonal iff M t M ,I . (106) Theorem 30: The set of orthogonal 2×2 matrices with determinant 1 together with matrix multiplication form a the special orthogonal group, which is written as SO ‘ 2, ë ’ . The set of orthogonal matrices together with matrix multiplication is also a group, the orthogonal group, which is written as O ‘ 2, ë ’ . SO ‘ 2, ë ’ is a subgroup of O ‘ 2, ë ’ . (107) Proposition 31: For two matrices A and B, (107) 1. ‘ A B’t ,B t A t 2. ‘ At ’ 1,‘ A 1 ’t ¤ ¤ 3. det ‘ A B’,det A4det B 4. det ‘ A t ’,det A 5. det ‘ At A’,det At4det A,det A4det A,‘det A’2 Fact 32: SO ‘2,ë’, • cos ¾ 1sin ¾ sin ¾ cos ¾ – }angle ¾  Def: Given a set G and an operation 2 : (113) G is a groupoid iff G is closed under 2 . G is a semigroup iff G is a groupoid and 2 is associative. G is a semigroup with identity iff G is a semigroup and has an identity under 2 . G is a group iff G is a semigroup and each element has an inverse under 2 . Def: A ring, written ‘R, 2 , ( ’ , consists of a nonempty set R and two opertaions such that (114) v ‘R, 2’ is an abelian group, v ‘R, ( ’ is a semigroup, and v the semigroup operation, ( , distributes over the group operation, 2 . Proposition 33: Let ‘R, 0 ,4’ be a ring. (115) 1. 04a, a40,0 } a V R 2. ‘1a’4b,a4‘1b’,1‘a4b’ } a, bV R 3. ‘1a’4‘1b’,a4b } a, bV R Def: A ring with identity is a ring that contains an indentity under the second operation (the multiplicative operation). (117) Def: A commutative ring is a ring where the second operation is commutative. (117) Def: A subring is a nonempty subset S of a ring ‘R, 0 ,4’ such that ‘S, 0 ,4’ is a ring (under the same operations as R.) (119) Proposition 34: To prove that ‘S, 0 , ( ’ is a subring of ‘R, 0 ,4’ we need to prove that (119) 1. S P R (set containment) 2. } a, bV S ‘a0b’V S (closure under additive operation) 3. } a, bV S ‘a4b’V S (closer under multiplicative operation) 4. } aV S ‘1a’V S (additive inverses exist in S)
Def: A ring ‘R, 0 ,4’ has zero divisors iff Y a, b V R such that a/0, b/0, and a4b,0 . (120) Def: In a ring ‘R, 0 ,4’ with identity, an element r is invertible iff Y r 1 V R such that r4r 1, r 14r ,1 ¥ ¥ ¥ (the multiplicative identity). (121) Proposition 35: Let R be the set of all invertible elements of R. If ‘R, 0 ,4’ is a ring with identity ¦ then ‘R ,4’ is a group, known as the group of invertible elements. (121) § Proposition 36: Let ‘R, 0 ,4’ be a ring with identity such that R/0 . The elements 0 and 1 are distinct. (122) Proposition 37: A ring ‘R, 0 ,4’ has no zero divisors iff the cancellation law for multiplication holds. (123) Corollary 38: Let ‘R, 0 ,4’ be a ring with identity which has no zero divisors. The only solutions to x 2, x in the ring are x,0 and x,1 . (123) Def: An integral domain is a commutative ring with identity which has no zero divisors. (124) Def: A field ‘F, 0 ,4’ is a set F together with two operations such that (125) v ‘F, 0 ’ is an abelian group, v ‘F10,4’ is an abelian group, and v 4 distributes over 0 . In other words, a field is a commutative ring with identity in which every nonzero element has an inverse. Back to intro and comments page: http://www.glump.net/archive/000024.php Back to home page: http://www.glump.net/

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Abstract Algebra Cheat Sheet

  • 1. Abstract Algebra Cheat Sheet 16 December 2002 By Brendan Kidwell, based on Dr. Ward Heilman’s notes for his Abstract Algebra class. Notes: Where applicable, page numbers are listed in parentheses at the end of a note. Def: A group is a nonempty set G together with a binary operation 2 on G3G satisfying the following four properties: 1. G is closed under the operation 2 . 2. The operation 2 is associative. 3. G contains an identity element, e, for the operation 2 . 4. Each element in G has an inverse in G under the operation 2 . Proposition 1: A group has exactly one identity element. Proposition 2: Each element of a group has exactly one inverse element. Proposition 3: ‘a2b’ 1,b 12a 1 } a, bV‘G, 2’ .       Proposition 4: ‘a 1 ’ 1,a } aV‘G, 2’ .     Proposition 5: ‘én , 0 n ’ is a group } nVè . Proposition 6: In a group table, every element occurs exactly once in each row and exactly once in each column. Def: The order of a group ‘G, 2’ is the number of elements in the set G. (Written as bGb .) (36) Def: A dihedral group of order 2n is the set of symmetric transformations of a regular n -gon . (Written as D n .) (36) Def: An abelian (or commutative) group has the property that a2b,b2a } a, bV‘G, 2’ . (37) Def: ‘H, 2’ is a subgroup of ‘G, 2’ iff H VG and ‘H, 2’ is a group under the same operation. (37) To show that ‘H, 2’ is a subgroup, show that H VG and then show closure and existence of inverses. Lagrange’s Theorem: Let ‘H, 2’ be a subgroup of a finite group, ‘G, 2’ . bHb divides bGb . Def: “ a ”,a 0 , a1 , a 1 , a 2 , a 2 , a3 , a 3 e is the cyclic subgroup generated by a.       Def: The order of an element, a, is the order of “ a” . Def: A cyclic group is a group that can be generated entirely by repeatedly combining a single element with itself. In other words, if for a cyclic group G,“ a” , then a is the generator of G. Def: Prime Order Proposition. For every prime p, there is exactly one group of order p. Proposition 8: Cancellation Laws. Let a, b, cV‘G, 2’ . 1. ‘a2b,a2c’ l‘b,c’ 2. ‘b2a,c2a’ l‘b,c’ 3. If G is abelian, ‘a2b,c2a’ l‘b,c’ Proposition 9: The only solution to a2a,a is a,e . Proposition 10: Let a, b V G . If a2b/b2a , then e, a, b, a2b, b2a are all distinct elements. (50)
  • 2. Proposition 11: Any non-abelian group has at least six elements. (51) Def: The center of a group is Z ‘G’,all g VG such that ‘ g2a,a2g } aVG ’ . Proposition 12: ‘Z ‘G’, 2’ is a subgroup of G. (52) Def: Two integers, a and b, are relatively prime iff gcd ‘a, b’,1 . (54) Def. } nVè , the set of units of n, U ‘n’ , is the set of all natural numbers relatively prime to n. (54) Proposition 13: } nVè , ‘U ‘n’,4n ’ is a group. (54) Def: For any set S and subsets A, B VS , the symmetric difference of A and B (written as A´ B ) is the set of all elements that are in A or B, but are not in both A and B. In other words, A ´ B ,‘ A1 B ’=‘ B 1 A ’ . (55) Def: The power set of S (written as P‘S ’ ) is the set of all subsets of S, including Ž and the original set S. (55) Proposition 14: For any nonempty set S, ‘ P‘S ’, ´ ’ is a group. (55) Def: Let ‘G, 2’ and ‘K, ( ’ be two groups. Let f be a function from G to K. f is a homomorphism (or operation preserving function) from ‘G, 2’ to ‘K, ( ’ iff } a, bVG f ‘a2b’, f ‘a’( f ‘b’ . (59) Proposition 15: Let f: G l K be a homomorphism. Let e be the identity of ‘G, 2’ and e ’ be the identity of ‘K, ( ’ . (60) 1. f ‘e’,e ’ 2. f ‘ g 1 ’,‘ f ‘ g ’’ 1 } g V G ¡ ¡ 3. f ‘g n ’,‘ f ‘g’’n } nV é Def: Given nonempty sets S and T, with x, y V S , and a function f: S lT (63) 1. f is a one-to-one (1-1) function iff ‘ x/ y ’ l ‘ f ‘x’/ f ‘ y’’ . 2. f is onto T iff } z VT YxV S such that f ‘x’, z . Proposition 16: Let f: S lT be an onto function. (65) 1. f ‘ f 1 ‘V ’ ’,V }V PT ¢ 2. W P f ‘ f 1 ‘W ’’ } W PS ¢ Proposition 17: Let f be a homomorphism from ‘G, 2’ to ‘K, ( ’ . (68) 1. If ‘H, 2’ is a subgroup of ‘G, 2’ , then ‘ f ‘H ’, ( ’ is a subgroup of ‘K, ( ’ . 2. If ‘L, ( ’ is a subgroup of ‘K, ( ’ , then ‘ f 1 ‘L’, 2 ’ is a subgroup of ‘G, 2’ . ¢ Def: (Using the previous example,) the image of H under f is f ‘H ’ . The inverse image of L under f is f 1 ‘L’ . (68) ¢ Proposition 18: Let f be a homomorphism from ‘G, 2’ to ‘K, ( ’ . f is one-to-one iff ker ‘ f ’,e  . (72) Def: Two groups, ‘G, 2’ and ‘K, ( ’ , are isomorphic iff there exists a one-to-one homomorphism f from ‘G, 2’ onto ‘K, ( ’ —that is, f ‘G’,K . In this case, f is called an isomorphism or isomorphic mapping. (73) Proposition 19: Every finite cyclic group of order n is isomorphic to ‘é n, 0 n ’ and every infinite cyclic group is isomorphic to ‘é , 0 ’ . (75) Proposition 20: Every subgroup of a cyclic group is cyclic. (76)
  • 3. Theorem: If G is a finite group, p is a prime, and p k is the largest power of p which divides bGb , then G has a subgroup of order pk . Def: A permutation is a one-to-one and onto function from a set to itself. (77) Note: See pages 78 and 81 for examples of how to notate permutations. Def: The set of permutations on 1, 2, 3,e, n is written as S n . (79) Theorem 21: The set of all permutations together with composition, ‘S n, ( ’ , is a nonabelian group } nB3 . (79) Theorem 22: The set of all permutations on a set S (its symmetries), together with composition, ‘Sym S, ( ’ , is a group. (80) Theorem 23 (Cayley’s Theorem): Every group is isomorphic to a group of permutations. (82) Proposition 24: Every permutation can be written as a product of disjoint cycles in permutation notation. (86) Def: The length of a cycle in a permutation is the number of distinct objects in it. A cycle of length 2 is a transposition. (86) Proposition 25: Every cycle can be written as a product of transpositions (not necessarily distinct). (87) Def: A permutation is even (or odd) if it can be written as a product of an even (or odd) number of transpositions. (88) Def: The subset of S n which consists of all the even permutations of S n is called the alternating group on n and is written as An . (90) Def: Matrix multiplication, which is not commutative, is the standard way to combine matrices. To multiply a 2×2 matrix: (102) • –• – • a b e f c d g h , a e0b g a f 0b h c e0d g c f 0d h – Notes: A 2×2 matrix can be found to represent any linear transformation. The special matrix M, •cos ¾ 1sin ¾ sin ¾ cos ¾ – when mulpilied on the left with a YHFWRU LQ ë 2 ZLOO URWDWH LW FRXQWHUFORFNZLVH by the amount ¾ : M X initial , X rotated . (100) Def: The inverse under multiplication of a 2×2 matrix is computed as follows: (103) • – d 1b 1 • – a b a d1b c a d1b c £ , c d 1c a a d1b c a d1b c Def: The determinant of a 2×2 matrix is computed as follows: (104) det ‘• –’ a b c d ,a d1b c Def: A matrix is invertible iff its determinant is nonzero. (104) Theorem 29: The set of all invertible 2×2 made from elements of ë , together with matrix multiplication, forms a group, called the general linear group, which is written as GL‘2, ë’ . (105)
  • 4. Def: The special linear group is the group of 2×2 matrices with determinants of 1, written as SL‘2, ë’ . (106) Def: To get the transpose of a matrix, swap each element a i, j with the one on the opposite side of the main diagonal, a j, i . The transpose of a matrix M is written M t . (106) Def: A matrix M is orthogonal iff M t M ,I . (106) Theorem 30: The set of orthogonal 2×2 matrices with determinant 1 together with matrix multiplication form a the special orthogonal group, which is written as SO ‘ 2, ë ’ . The set of orthogonal matrices together with matrix multiplication is also a group, the orthogonal group, which is written as O ‘ 2, ë ’ . SO ‘ 2, ë ’ is a subgroup of O ‘ 2, ë ’ . (107) Proposition 31: For two matrices A and B, (107) 1. ‘ A B’t ,B t A t 2. ‘ At ’ 1,‘ A 1 ’t ¤ ¤ 3. det ‘ A B’,det A4det B 4. det ‘ A t ’,det A 5. det ‘ At A’,det At4det A,det A4det A,‘det A’2 Fact 32: SO ‘2,ë’, • cos ¾ 1sin ¾ sin ¾ cos ¾ – }angle ¾  Def: Given a set G and an operation 2 : (113) G is a groupoid iff G is closed under 2 . G is a semigroup iff G is a groupoid and 2 is associative. G is a semigroup with identity iff G is a semigroup and has an identity under 2 . G is a group iff G is a semigroup and each element has an inverse under 2 . Def: A ring, written ‘R, 2 , ( ’ , consists of a nonempty set R and two opertaions such that (114) v ‘R, 2’ is an abelian group, v ‘R, ( ’ is a semigroup, and v the semigroup operation, ( , distributes over the group operation, 2 . Proposition 33: Let ‘R, 0 ,4’ be a ring. (115) 1. 04a, a40,0 } a V R 2. ‘1a’4b,a4‘1b’,1‘a4b’ } a, bV R 3. ‘1a’4‘1b’,a4b } a, bV R Def: A ring with identity is a ring that contains an indentity under the second operation (the multiplicative operation). (117) Def: A commutative ring is a ring where the second operation is commutative. (117) Def: A subring is a nonempty subset S of a ring ‘R, 0 ,4’ such that ‘S, 0 ,4’ is a ring (under the same operations as R.) (119) Proposition 34: To prove that ‘S, 0 , ( ’ is a subring of ‘R, 0 ,4’ we need to prove that (119) 1. S P R (set containment) 2. } a, bV S ‘a0b’V S (closure under additive operation) 3. } a, bV S ‘a4b’V S (closer under multiplicative operation) 4. } aV S ‘1a’V S (additive inverses exist in S)
  • 5. Def: A ring ‘R, 0 ,4’ has zero divisors iff Y a, b V R such that a/0, b/0, and a4b,0 . (120) Def: In a ring ‘R, 0 ,4’ with identity, an element r is invertible iff Y r 1 V R such that r4r 1, r 14r ,1 ¥ ¥ ¥ (the multiplicative identity). (121) Proposition 35: Let R be the set of all invertible elements of R. If ‘R, 0 ,4’ is a ring with identity ¦ then ‘R ,4’ is a group, known as the group of invertible elements. (121) § Proposition 36: Let ‘R, 0 ,4’ be a ring with identity such that R/0 . The elements 0 and 1 are distinct. (122) Proposition 37: A ring ‘R, 0 ,4’ has no zero divisors iff the cancellation law for multiplication holds. (123) Corollary 38: Let ‘R, 0 ,4’ be a ring with identity which has no zero divisors. The only solutions to x 2, x in the ring are x,0 and x,1 . (123) Def: An integral domain is a commutative ring with identity which has no zero divisors. (124) Def: A field ‘F, 0 ,4’ is a set F together with two operations such that (125) v ‘F, 0 ’ is an abelian group, v ‘F10,4’ is an abelian group, and v 4 distributes over 0 . In other words, a field is a commutative ring with identity in which every nonzero element has an inverse. Back to intro and comments page: http://www.glump.net/archive/000024.php Back to home page: http://www.glump.net/