Subgroups - Linear Algebra - Lecture Notes | MATH 214 | Study notes Linear Algebra

Section 5: Subgroups

In the last section, we learned that a nonempty subset Sof a group Gwas a “subgroup” iff it

was closed under the operation in Gand under inverses. The text wisely points out that a subset

which is a group need not be a subgroup, because the operation may be differqent. For example,

(IQ,+) and (IQ+,·) are both groups, and IQ+⊂IQ, but IQ+is not called a subgroup of IQ.

We have already seen that, for any element xof any group G,hxiis a subgroup of G. In

particular, {e}is a subgroup of any group (the “trivial subgroup”); and of course any group is a

subgroup of itself. Here are a few other examples:

Examples:

•ZZ is a subroup (under addition) of IQ, which is a subgroup of IR, which is a subgroup of IC.

•IQ+is a subgroup under multiplication of IQ − {0}, which is a subgroup of IR − {0}, which is a

subgroup of IC − {0}. Another subgroup of IR − {0}is IR+, and of course IR+∩(IQ − {0}) = IQ+.

We will soon see that any intersection of subgroups is another subgroup.

•Any subspace of any vector space is a subgroup under addition, as well as being closed under

scalar multiplication, as we learn in Math 214. In particular, a plane through the origin of

IR3is a subgroup of IR3under addition.

•GL(n, IR) has many subgroups. One of the best known is SL(n, IR) = {A∈GL(n,IR) :

det(A)=1}, called the the special linear group of degree n. (Verify this is a subgroup. The

mathematician Serge Lang has written a book titled SL(2,IR).) The set of upper triangular

matrices is an additive subgroup of Mn(IR), and the set of upper triangular matrices with

no zeros on the main diagonal is a multiplicative subgroup of GL(n, IR). Replace “upper

triangular” (both times) in the last sentence with “lower triangular” or by “diagonal”, and

the sentence remains true.

•The Klein Four-Group V={e,a, b, c}is so small that its subgroups aren’t very interesting,

but at least it is easy to write them all down: hei,hai,hbi,hci, V . A “subgroup diagram”

or “subgroup lattice” displays their containment relationships: a line angling up means the

higher contains the lower as a subgroup.

•The text introduces the group Q8={I, J, K, L, −I , −J, −K, −L}of unit quaternions, where

J2=K2=L2=−I. The multiplication could have been determined by a table; but as

the text points out, it would be necessary to check associativity, so it uses the connection to

matrices in GL(2,IC). (It also could have been done in GL(4,IR), but the text avoids matrices

that large.) The diagram below to the left is a mnemonic (memory aid) for the operation:

clockwise is a positive product and counterclockwise is negative: JK =L, but KJ =−L,

etc. The diagram below to the right is the subgroup lattice:

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Section 5: Subgroups

In the last section, we learned that a nonempty subset S of a group G was a “subgroup” iff it was closed under the operation in G and under inverses. The text wisely points out that a subset which is a group need not be a subgroup, because the operation may be differqent. For example, (QI, +) and (QI+, ·) are both groups, and QI+^ ⊂ QI, but QI+^ is not called a subgroup of QI. We have already seen that, for any element x of any group G, 〈x〉 is a subgroup of G. In particular, {e} is a subgroup of any group (the “trivial subgroup”); and of course any group is a subgroup of itself. Here are a few other examples:

Examples:

ZZ is a subroup (under addition) of QI, which is a subgroup of IR, which is a subgroup of CI.
QI+^ is a subgroup under multiplication of QI − { 0 }, which is a subgroup of IR − { 0 }, which is a subgroup of CI − { 0 }. Another subgroup of IR − { 0 } is IR+, and of course IR+^ ∩ (QI − { 0 }) = QI+. We will soon see that any intersection of subgroups is another subgroup.
Any subspace of any vector space is a subgroup under addition, as well as being closed under scalar multiplication, as we learn in Math 214. In particular, a plane through the origin of IR^3 is a subgroup of IR^3 under addition.
GL(n, IR) has many subgroups. One of the best known is SL(n, IR) = {A ∈ GL(n, IR) : det(A) = 1}, called the the special linear group of degree n. (Verify this is a subgroup. The mathematician Serge Lang has written a book titled SL(2, IR).) The set of upper triangular matrices is an additive subgroup of Mn(IR), and the set of upper triangular matrices with no zeros on the main diagonal is a multiplicative subgroup of GL(n, IR). Replace “upper triangular” (both times) in the last sentence with “lower triangular” or by “diagonal”, and the sentence remains true.
The Klein Four-Group V = {e, a, b, c} is so small that its subgroups aren’t very interesting, but at least it is easy to write them all down: 〈e〉, 〈a〉, 〈b〉, 〈c〉, V. A “subgroup diagram” or “subgroup lattice” displays their containment relationships: a line angling up means the higher contains the lower as a subgroup.
The text introduces the group Q 8 = {I, J, K, L, −I, −J, −K, −L} of unit quaternions, where J^2 = K^2 = L^2 = −I. The multiplication could have been determined by a table; but as the text points out, it would be necessary to check associativity, so it uses the connection to matrices in GL(2, CI). (It also could have been done in GL(4, IR), but the text avoids matrices that large.) The diagram below to the left is a mnemonic (memory aid) for the operation: clockwise is a positive product and counterclockwise is negative: JK = L, but KJ = −L, etc. The diagram below to the right is the subgroup lattice:

I saw this group first in the context of the “algebra of quaternions”, in the more common notation 1 for I, i for J, j for K and k for L. The algebra of quaternions is the set of expressions a + bi + cj + dk , where a, b, c, d ∈ IR. It is an extension of CI over IR. Our text would deal with it as a “subalgebra of M 2 × 2 (CI) over IR”.

The symmetric group on n letters, Sn, i.e., the set of one-to-one functions from { 1 , 2 ,... , n} onto itself, a group under composition, has a subgroup: the set of all elements f of Sn for which f (n) = n — i.e., the functions that don’t move n. It is fairly clear that this subgroup “behaves exactly like” Sn− 1. And there is nothing special here about the set { 1 , 2 ,... , n} and the subset {n}: For any set X and subset Y of X, the symmetric group S(X) has a subgroup consisting of the set of all elements f of S(X) for which f (y) = y for every y in Y , which “behaves exactly like” S(X − Y ). The subset of all elements f of S(X) for which f (y) ∈ Y for every y in Y and f (z) ∈ X − Y for every z ∈ X − Y is another subgroup of S(X). (If Y is finite, then we don’t need the extra part about z’s in X − Y , because if f takes all of Y to itself, then f (Y ) is all of Y , so z’s outside of Y must go to elements outside of Y. But if Y is infinite, then a function may take Y into Y but not onto it, so its inverse would not take Y into Y. Example: The “add one” function f (x) = x + 1 is an element of S(ZZ) for which f (ZZ+) ⊆ ZZ+; but its inverse, the “subtract one” function, takes 1 to 0, outside of ZZ+. So the subset of S(ZZ) consisting of functions f for which f (ZZ+) ⊆ ZZ+^ is not a subgroup of S(ZZ), because it is not closed under inverses.)
Let G be any group and x be a fixed element of G. Then Z(x) = {g ∈ G : gx = xg}, the set of all elements that “commute with” x, is easily checked to be a subgroup of G, called the centralizer of x. Again, there is nothing special about a single-element set {x}: For any subset X of G, the “centralizer of X”, Z(X) = {g ∈ G : gx = xg ∀x ∈ X}, is a subgroup of G. (We could make this a corollary of the result below that an intersection of a family of subgroups is again a subgroup, because Z(X) = ⋂ {Z(x) : x ∈ X}.) In particular, the centralizer of G itself, Z(G), the set of all elements of G that commute with every element of G, is a subgroup, called the center of G. Some examples of centers and centralizers:

∗ If G is abelian, then of course Z(G) = G and for each element x of G, Z(x) = G. More generally, if x ∈ Z(G), then Z(x) = G. ∗ In the group Q 8 of unit quaternions, Z(Q 8 ) = 〈−I〉, because none of the other 6 elements commute with everything. But Z(〈J〉) = 〈J〉; the powers of J commute with J, but none of the other four elements of Q 8 do.

was chosen so that xk^ was the smallest positive power of x in H, so r can’t be positive, i.e., r = 0. Thus, xm^ = (xk)d^ ∈ 〈xk〉, and so H ⊆ 〈xk〉. (ii) Clearly the sets of powers of xk^ and x−k^ are the same set, so the given list includes all the subgroups of G; so we only have to show that they are distinct if G is infinite cyclic. The text leaves this as an exercise for the reader, so I will, too. (iii) Now we are supposing that G is finite of order n. From the proof given in (i), any nontrivial subgroup H of G has the form 〈xk〉 where k is the smallest positive integer for which xk^ ∈ H. We want to show that this k is a divisor of n: We know xn^ = e ∈ H, and long-dividing n by k shows, just as in (i), that k|n, say n = kd. We proved in Section 4 that o(xk) = n/ gcd(n, k) = n/k, so 〈xk〉 is a subgroup of G of order n/k. Thus, every subgroup of G has order a divisor n/k = d of n and is generated by xk^ = xn/d. Conversely, if d is a divisor of n, then we also proved in Section 4 that xn/d^ is an element of order n/ gcd(n, (n/d)) = n/(n/d) = d, so 〈xn/d〉 is a subgroup of order d.//

Cor: In a finite cyclic group G = 〈x〉 of order n, 〈xk〉 = 〈xgcd(n,k)〉. In particular, 〈xr〉 = 〈xs〉 iff gcd(n, r) = gcd(n, s).

Pf: For the first equality, we need to show that each of xk^ and xgcd(n,k)^ is a power of the other. Because k is a multiple of gcd(n, k), xk^ is a power of xgcd(n,k); and because gcd(n, k) = nr + ks for some r, s in ZZ, we have xgcd(n,k)^ = (xn)r(xk)s^ = (xk)s. Thus, 〈xk〉 = 〈xgcd(n,k)〉. It follows that, if gcd(n, r) = gcd(n, s), then 〈xr〉 = 〈xs〉. Conversely, if 〈xr〉 = 〈xs〉, then 〈xgcd(n,r)〉 = 〈xgcd(n,s)〉, and because the subgroups are equal and the exponents are factors of n, the exponents gcd(n, r) and gcd(n, r) are also equal.//

The text also includes the following useful fact: If S is a finite nonempty subset of a group G and S is closed under the operation, then S is also closed under inverses and hence is a subgroup of G. The proof is simple: Let x ∈ S; then because S is finite and closed under the operation, the powers of x cannot all be different, say xp^ = xq^ where p < q. We get xq−p^ = e, so x has finite order, and its inverse is a positive power of it, so x−^1 ∈ S.

Subgroups - Linear Algebra - Lecture Notes | MATH 214, Study notes of Linear Algebra

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