Ch. 5: Subgroups & Product Groups!

Ch. 5: Subgroups & Product Groups!

DEFINITION: A group is a set (denoted G) with an algebraic operation (denoted •) that satisfies the following properties: (1) G has an “identity” (denoted I) which has no effect on other members; that is, A•I= A and I•A = A for all members, A, of G. (2) Each member, A, of G has an inverse in G (usually denoted A–1), which combines with it in either order to give the identity: A•A–1 = I and A–1•A = I. (3) The associative property holds: (A•B)•C = A•(B•C) for all triples A,B,C of members of G. Here are three examples of groups: (1) The numbers (under addition). The identity is 0 The inverse of 35 is -35. (2) The numbers (under multiplication). The identity is 1 The inverse of 35 is 1/35. (3) The 8 symmetries of a square (under composition). The identity is I The inverse of R90 is R270. The inverse of H is H.

D4 = {I, R90, R180, R270, H, V, D, D’} includes: { H, V, D, D’ } “the flips” {I, R90, R180, R270} “the rotations” Which color forms a self-contained group on its own?

D4 = {I, R90, R180, R270, H, V, D, D’} includes: { H, V, D, D’ } “the flips” {I, R90, R180, R270} “the rotations” “The rotations” is a group on its own (called C4); This is a Cayley table “the flips” is NOT a group on its own; This is NOT a Cayley table

D4 = {I, R90, R180, R270, H, V, D, D’} includes: { H, V, D, D’ } “the flips” {I, R90, R180, R270} “the rotations” If you color some of the members of a group, under what conditions will the colored collection form a self-contained group on its own?

D4 = {I, R90, R180, R270, H, V, D, D’} includes: If you list all the members of G, and color some of them red, then the red ones form a subgroup(a self-contained group on its own) if… (1) You colored the identity red. (2) When any pair of red members are combined, the answer is red. (3) The inverse of any red member is red.

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. If you list all the members of G, and color some of them red, then the red ones form a subgroup(a self-contained group on its own) if… (1) You colored the identity red. (2) When any pair of red members are combined, the answer is red. (3) The inverse of any red member is red.

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. EXAMPLE: In D4 = {I, R90, R180, R270, H, V, D, D’} Do the red members {I, R180, H, V} form a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. EXAMPLE: In D4 = {I, R90, R180, R270, H, V, D, D’} Do the red members {I, R180, H, V} form a subgroup? YES! This subgroup is called D2. Think of it as the symmetry group of the “striped square”

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. W WWWWWWWWWWWWWWW In the symmetry group of the W-border pattern, Do the translations alone form a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. W WWWWWWWWWWWWWWW In the symmetry group of the W-border pattern, Do the translations alone form a subgroup? YES!

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. W WWWWWWWWWWWWWWW In the symmetry group of the W-border pattern, Do the translations alone form a subgroup? What about the vertical flips alone? YES!

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. W WWWWWWWWWWWWWWW In the symmetry group of the W-border pattern, Do the translations alone form a subgroup? What about the vertical flips alone? YES! NO!

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. In the additive group of integers, Z = {…,-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, …}, Do the even numbers, E = {…, -8, -6, -4, -2, 0, 2, 4, 6, 8, …} form a subgroup? What about the odd numbers?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. In the additive group of integers, Z = {…,-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, …}, Do the even numbers, E = {…, -8, -6, -4, -2, 0, 2, 4, 6, 8, …} form a subgroup? What about the odd numbers? YES! NO!

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. In the cyclic group, C10= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. In the cyclic group, C10= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup? YES!

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. In the cyclic group, C10= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup? YES! In the cyclic group, C9= {0, 1, 2, 3, 4, 5, 6, 7, 8} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. In the cyclic group, C10= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup? YES! In the cyclic group, C9= {0, 1, 2, 3, 4, 5, 6, 7, 8} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup? NO!

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. C12= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. C12= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} <2> = “the subgroup of C12 generated by 2” Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. C12= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} Oops, your little brother colored the 5 with his red crayon. To fix it, what else must you color red to make it a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. C12= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} <5> = “the subgroup of C12 generated by 5” Oops, your little brother colored the 5 with his red crayon. To fix it, what else must you color red to make it a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. Z= {…-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. Z= {…-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} <2> = “the subgroup of Zgenerated by 2” Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup?

DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then A•B must be in H. (Inverses) The inverse of anything in H must be in H. Z= {…-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} <2> = “the subgroup of Zgenerated by 2” DEFINITION: If G is a group, and A is a member of G, then, <A> = {…, A–1•A–1•A–1, A–1•A–1, A–1, I, A, A•A, A•A•A,…} is called the subgroup of G generated by A. (it contains A and A–1 combined with themselves any number of times.)

DEFINITION: If G is a group, and A is a member of G, then, <A> = {…, A–1•A–1•A–1, A–1•A–1, A–1, I, A, A•A, A•A•A,…} is called the subgroup of G generated by A. (it contains A and A–1 combined with themselves any number of times.) D4= {I, R90, R180, R270, H, V, D, D’} Find the subgroup of D4generated by each of its 8 members. <I> = ? < R90 > = ? <R180 > = ? < R270 > = ? <H> = ? <V> = ? <D> = ? <D’> = ?

DEFINITION: If G is a group, and A is a member of G, then, <A> = {…, A–1•A–1•A–1, A–1•A–1, A–1, I, A, A•A, A•A•A,…} is called the subgroup of G generated by A. (it contains A and A–1 combined with themselves any number of times.) D4= {I, R90, R180, R270, H, V, D, D’} Find the subgroup of D4generated by each of its 8 members. <I> = { I } < R90 > = { I, R90, R180, R270} < R180 > = { I, R180} < R270 > = { I, R270, R180, R90} <H> = { I, H } <V> = { I, V } <D> = { I, D } <D’> = { I, D’ } the same

DEFINITION: If G is a group, and A is a member of G, then, <A> = {…, A–1•A–1•A–1, A–1•A–1, A–1, I, A, A•A, A•A•A,…} is called the subgroup of G generated by A. (it contains A and A–1 combined with themselves any number of times.) D4= {I, R90, R180, R270, H, V, D, D’} Find the subgroup of D4generated by each of its 8 members. THEOREM: If G is a finite group, and A is a member of G, then <A> = { I, A, A•A, A•A•A, A•A•A•A, …} (this list starts repeating as soon as one of these expressions equals I, and not before). (you don’t need to worry about inverses)

DEFINITION: If G is a group, and A is a member of G, then, <A> = {…, A–1•A–1•A–1, A–1•A–1, A–1, I, A, A•A, A•A•A,…} is called the subgroup of G generated by A. (it contains A and A–1 combined with themselves any number of times.) D4= {I, R90, R180, R270, H, V, D, D’} Find the subgroup of D4generated by each of its 8 members. THEOREM: If G is a finite group, and A is a member of G, then <A> = { I, A, A•A, A•A•A, A•A•A•A, …} (this list starts repeating as soon as one of these expressions equals I, and not before). DEFINITION: If A is a member of a finite group, then the order of A is the size of the subgroup <A>. Find order of each member of D4.

In the group C10= { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }, find: <2> = <3> = <4> = <5> = What is the order of each member of this group?

In the group C10= { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }, find: <2> = {0, 2, 4, 6, 8} (the evens) <3> = {0, 3, 6, 9, 2, 5, 8, 1, 4, 7} (all of C10) <4> = {0, 4, 8, 2, 6} (the evens – the same as <2>) <5> = {0, 5}. 2 has order 5 3 has order 10 4 has order 5 5 has order 2

Product Groups

G1×G2means all of the possible ways of pairing together a member of the first group with a member of the second group (wrapped in parentheses and separated by a comma) It is called the product of G1 & G2 EXAMPLE: If G1= { A, B, C } and G2= { 1, 2, 3, 4 }, then: G1×G2 = { (A,1), (A,2), (A,3), (A,4), (B,1), (B,2), (B,3), (B,4), (C,1), (C,2), (C,3), (C,4) }

G1×G2means all of the possible ways of pairing together a member of the first group with a member of the second group (wrapped in parentheses and separated by a comma) It is called the product of G1 & G2 EXAMPLE: If G1= { A, B, C } and G2= { 1, 2, 3, 4 }, then: G1×G2 = { (A,1), (A,2), (A,3), (A,4), (B,1), (B,2), (B,3), (B,4), (C,1), (C,2), (C,3), (C,4) } In this example G1×G2 has 12 members , because the members of can be arranged into a 3x4 table. GENERAL RULE: The size of the product of two finite groups equals the product of their sizes

The Cayley table of G1×G2 Just combine the G1-part and the G2-part separately! EXAMPLE: In D4×Z, compute (H,6) • (V,8) = ( ??, ?? ) Recall: Z= {…, –3, –2, –1, 0, 1, 2, 3, …} = the additive group of integers.

The Cayley table of G1×G2 Just combine the G1-part and the G2-part separately! EXAMPLE: In D4×Z, compute (H,6) • (V,8) = ( R180, 14 ) H*V = R180in D4 6 + 8 = 14 in Z

The product of C3= { 0, 1, 2 } and C2= { 0, 1} has the following six members: C3×C2 = {(0,0), (0,1), (1,0), (1,1), (2,0), (2,1)}. What does the Cayley table look like?

The product of C3= { 0, 1, 2 } and C2= { 0, 1} has the following six members: C3×C2 = {(0,0), (0,1), (1,0), (1,1), (2,0), (2,1)}. Fill in the red and green separately: (1,1) • (1,0) = (2,1) because… 1+1 = 2 in C3 and……… 1+0 = 1 in C2.

The product of C3= { 0, 1, 2 } and C2= { 0, 1} has the following six members: C3×C2 = {(0,0), (0,1), (1,0), (1,1), (2,0), (2,1)}. The Cayley table looks like this: (1,1) • (1,0) = (2,1) because… 1+1 = 2 in C3 and……… 1+0 = 1 in C2.

Some symmetry groups are product groups in disguise. Explain why D2 = the symmetry group of this cross …is isomorphic to C2×C2.

Some symmetry groups are product groups in disguise. Explain why D2 = the symmetry group of this cross …is isomorphic to C2×C2. Recall that C2 = {0,1} = the symmetry group of an oriented 2-gon. Cayley table: or… Another reasonable notation is: 0 = “N” = “No rotate” 1 = “Y” = “Yes rotate”

Some symmetry groups are product groups in disguise. Explain why D2 = the symmetry group of this cross …is isomorphic to C2×C2. An isomorphism: I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R180 ↔ (1,1) Easy to check that this dictionary translates a correct Cayleyinto a correct Cayley table…

Some symmetry groups are product groups in disguise. Explain why D2 = the symmetry group of this cross …is isomorphic to C2×C2. An isomorphism: I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R180 ↔ (1,1) Easy to check that this dictionary translates a correct Cayleyinto a correct Cayley table… …but what is the visual meaning?

Some symmetry groups are product groups in disguise. Explain why D2 = the symmetry group of this cross …is isomorphic to C2×C2. An isomorphism: I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R180 ↔ (1,1) Doesn’t exchange any ends Exchanges red ends but not green ends Exchanges green ends but not red ends Exchanges red ends and green ends Are the ends of the green rectangle exchanged? (0=NO, 1=YES) Are the ends of the red rectangle exchanged? (0=NO, 1=YES)

Some symmetry groups are product groups in disguise. Explain why D2 = the symmetry group of this cross …is isomorphic to C2×C2. An isomorphism: I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R180 ↔ (1,1) Which viewpoint is simpler… V*H=R180 Exchanging red ends and then greenends results in both ends being exchanged.

Some symmetry groups are product groups in disguise. B BBBBBBBBBBBBBBBBBBBB Explain why the symmetry group of the B-border pattern …is isomorphic to Z×C2.

Some symmetry groups are product groups in disguise. B BBBBBBBBBBBBBBBBBBBB Explain why the symmetry group of the B-border pattern …is isomorphic to Z×C2. Recall: Z= {…, –3, –2, –1, 0, 1, 2, 3, …} = the additive group of integers.

Ch. 5: Subgroups & Product Groups!