4: Subgroups
( \newcommand{\kernel}{\mathrm{null}\,}\)
From now on, unless otherwise stated, G will denote a group whose binary operation is denoted by a⋅b or simply ab for a,b∈G. The identity of G will be denoted by e and the inverse of a∈G will be denoted by a−1. Sometimes, however, we may need to discuss groups whose operations are thought of as addition. In such cases we write a+b instead of ab. Also in this case, the identity is denoted by 0 and the inverse of a∈G is denoted by −a. Definitions and results given using multiplicative notation can always be translated to additive notation if necessary.
Let G be a group. A subgroup of G is a subset H of G which satisfies the following three conditions:
- e∈H.
- If a,b∈H, then ab∈H.
- If a∈H, then a−1∈H.
For convenience we sometimes write H≤G to mean that H is a subgroup of G.
Problem 4.1 Translate the above definition into additive notation.
If H is a subgroup of G, then the binary operation on G when restricted to H is a binary operation on H. From the definition, one may easily show that a subgroup H is a group in its own right with respect to this binary operation. Many examples of groups may be obtained in this way. In fact, in a way we will make precise later, every finite group may be thought of as a subgroup of one of the groups Sn.
Problem 4.2 Prove that if G is any group, then
- {e}≤G.
- G≤G.
The subgroups {e} and G are said to be trivial subgroups of G.
Problem 4.3
(a) Determine which of the following subsets of S4 are subgroups of S4.
- H={ι,(1 2),(3 4),(1 2)(3 4)}
- K={ι,(1 2 3),(1 3 2)}
- J={ι,(1 2),(1 2 3)}
- L={σ∈S4 | σ(1)=1}.
(b) Determine which of the following subsets of Z12 are subgroups of Z12. (Here the binary operation is addition modulo 12.)
- A={0,3,6,9,}
- B={0,6}
- C={0,1,2,3,4,5}
(c) Determine which of the following subsets of Z are subgroups of Z. (Here the binary operation is addition.)
- U={5k |k ∈Z}
- V={5k+1 | k∈N}
- W={5k+1 | k∈Z}
Problem 4.4 Let SL(2,R)={A∈GL(2,R)|det(A)=1}. Prove that SL(2,R)≤GL(2,R).
SL(2,R) is called the Special Linear Group of Degree 2 over R
Problem 4.5 For n∈N, let An be the set of all even permutations in the group Sn. Show that An is a subgroup of Sn.
An is called the alternating group of degree n.
Problem 4.6 List the elements of An for n=1,2,3,4. Based on this try to guess the order of An for n>4.
Let a be an element of the group G. If there exists n∈N such that an=e we say that a has finite order. and we define o(a)=min{n∈N|an=e} If an≠e for all n∈N, we say that a has infinite order and we define o(a)=∞. In either case we call o(a) the order of a.
Note carefully the difference between the order of a group and the order of an element of a group. Some authors make matters worse by using the same notation for both concepts. Maybe by using different notation it will make it a little easier to distinguish the two concepts.
If n≥2, to prove that o(a)=n we must show that ai≠e for i=1,2,…,n−1 and an=e. Note also that a=e if and only if o(a)=1. So every element of a group other than e has order n≥2 or ∞.
Problem 4.7 Translate the above definition into additive notation. That is, define the order of an element of a group G with binary operation + and identity denoted by 0.
Problem 4.8 Find the order of each element of S3.
Problem 4.9 Find the order of a k-cycle when k=2,3,4,5. Guess the order of a k-cycle for arbitrary k.
Problem 4.10 Find the order of the following permutations:
(a) (1 2)(3 4 5)
(b) (1 2)(3 4)(5 6 7 8)
(c) (1 2)(3 4)(5 6 7 8)(9 10 11)
(d) Try to find a rule for computing the order of a product disjoint cycles in terms of the sizes of the cycles.
Problem 4.11 Find the order of each element of the group (Z6,+).
Problem 4.12 Find the order of each element of GL(2,Z2). [Recall that GL(2,Z2) is the group of all 2×2 matrices with entries in Z2 with non-zero determinant. Recall that Z2={0,1} and the operations are multiplication and addition modulo 2.]
Problem 4.13 Find the order of the element 2 in the group (R−{0},⋅). Are there any elements of finite order in this group?
Let a be an element of the group G. Define ⟨a⟩={ai:i∈Z}. We call ⟨a⟩ the subgroup of G generated by a.
Note that ⟨a⟩={…,a−3,a−2,a−1,a0,a1,a2,a3,…}. In particular, a=a1 and e=a0 are in ⟨a⟩.
Problem 4.14 Translate the above definition of ⟨a⟩ and the remark into additive notation.
For each a in the group G, ⟨a⟩ is a subgroup of G. ⟨a⟩ contains a and is the smallest subgroup of G that contains a. ◼
Proof As just noted e=a0∈⟨a⟩. Let an,am∈⟨a⟩. Since n+m∈Z it follows from Theorem 2.4 that anam=an+m∈⟨a⟩. Also from Theorem 2.4, if an∈⟨a⟩, since n(−1)=−n we have (an)−1=a−n∈⟨a⟩. This proves that ⟨a⟩ is a subgroup.
Since a=a1 it is clear that a∈⟨a⟩. If H is any subgroup of G that contains a, since H is closed under taking products and taking inverses, an∈⟨a⟩ for every n∈Z. So ⟨a⟩⊆H. That is, every subgroup of G that contains a also contains ⟨a⟩. This implies that ⟨a⟩ is the smallest subgroup of G that contains a.
Let G be a group and let a∈G. If o(a)=1, then ⟨a⟩={e}. If o(a)=n where n≥2, then ⟨a⟩={e,a,a2,…,an−1} and the elements e,a,a2,…,an−1 are distinct, that is, o(a)=|⟨a⟩|. ◼
Proof Assume that o(a)=n. The case n=1 is left to the reader. Suppose n≥2. We must prove two things.
- If i∈Z then ai∈{e,a,a2,…,an−1}.
- The elements e,a,a2,…,an−1 are distinct.
To establish 1 we note that if i is any integer we can write it in the form i=nq+r where r∈{0,1,…,n−1}. Here q is the quotient and r is the remainder when i is divided by n. Now using Theorem [Th2.3] we have ai=anq+r=anqar=(an)qar=eqar=ear=ar. This proves 1. To prove 2, assume that ai=aj where 0≤i<j≤n−1. It follows that aj−i=aj+(−i)=aja−i=aia−i=a0=e. But j−i is a positive integer less than n, so aj−i=e contradicts the fact that o(a)=n. So the assumption that ai=aj where 0≤i<j≤n−1 is false. This implies that 2 holds. It follows that ⟨a⟩ contains exactly n elements, that is, o(a)=|⟨a⟩|.
If G is a finite group, then every element of G has finite order. ◼
Proof Let a be any element of G. Consider the infinite list a1,a2,a3,…,ai,… of elements in G. Since G is finite, all the elements in the list cannot be different. So there must be positive integers i<j such that ai=aj. Since i<j, j−i is a positive integer. Then using Theorem 2.4 we have aj−i=aj+(−i)=aja−i=aia−i=a0=e. That is, an=e for the positive integer n=j−i. So a has finite order, which is what we wanted to prove.
Problem 4.15 For each choice of G and each given a∈G list all the elements of the subgroup ⟨a⟩ of G.
- G=S3, a=(1 2).
- G=S3, a=(1 2 3).
- G=S4, a=(1 2 3 4).
- G=S4, a=(1 2)(3 4).
- G=Z, a=5.
- G=Z, a=−1.
- G=Z15, a=5.
- G=Z15, a=1.
- G=GL(2,Z2), a=(1101).
- G=GL(2,R), a=(0−110).
Problem 4.16 Suppose a is an element of a group and o(a)=n. Prove that am=e if and only if n|m. [Hint: The Division Algorithm from Appendix C may be useful for the proof in one direction.]