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4: Subgroups

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From now on, unless otherwise stated, G will denote a group whose binary operation is denoted by ab or simply ab for a,bG. The identity of G will be denoted by e and the inverse of aG will be denoted by a1. 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 aG is denoted by a. Definitions and results given using multiplicative notation can always be translated to additive notation if necessary.

Definition 4.1:

Let G be a group. A subgroup of G is a subset H of G which satisfies the following three conditions:

  1. eH.
  2. If a,bH, then abH.
  3. If aH, then a1H.

For convenience we sometimes write HG to mean that H is a subgroup of G.

Problem 4.1 Translate the above definition into additive notation.

Remark

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

  1. {e}G.
  2. GG.

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.

  1. H={ι,(1 2),(3 4),(1 2)(3 4)}
  2. K={ι,(1 2 3),(1 3 2)}
  3. J={ι,(1 2),(1 2 3)}
  4. L={σS4 | σ(1)=1}.

(b) Determine which of the following subsets of Z12 are subgroups of Z12. (Here the binary operation is addition modulo 12.)

  1. A={0,3,6,9,}
  2. B={0,6}
  3. 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.)

  1. U={5k |k Z}
  2. V={5k+1 | kN}
  3. W={5k+1 | kZ}

Problem 4.4 Let SL(2,R)={AGL(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 nN, 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.

Definition 4.2:

Let a be an element of the group G. If there exists nN such that an=e we say that a has finite order. and we define o(a)=min{nN|an=e} If ane for all nN, 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 n2, to prove that o(a)=n we must show that aie for i=1,2,,n1 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 n2 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?

Definition 4.3:

Let a be an element of the group G. Define a={ai:iZ}. We call a the subgroup of G generated by a.

Remark

Note that a={,a3,a2,a1,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.

Theorem 4.1

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=a0a. Let an,ama. Since n+mZ it follows from Theorem 2.4 that anam=an+ma. Also from Theorem 2.4, if ana, since n(1)=n we have (an)1=ana. This proves that a is a subgroup.

Since a=a1 it is clear that aa. If H is any subgroup of G that contains a, since H is closed under taking products and taking inverses, ana for every nZ. So aH. 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.

Theorem 4.2

Let G be a group and let aG. If o(a)=1, then a={e}. If o(a)=n where n2, then a={e,a,a2,,an1} and the elements e,a,a2,,an1 are distinct, that is, o(a)=|a|.

Proof Assume that o(a)=n. The case n=1 is left to the reader. Suppose n2. We must prove two things.

  1. If iZ then ai{e,a,a2,,an1}.
  2. The elements e,a,a2,,an1 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,,n1}. 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 0i<jn1. It follows that aji=aj+(i)=ajai=aiai=a0=e. But ji is a positive integer less than n, so aji=e contradicts the fact that o(a)=n. So the assumption that ai=aj where 0i<jn1 is false. This implies that 2 holds. It follows that a contains exactly n elements, that is, o(a)=|a|.

Theorem 4.3

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, ji is a positive integer. Then using Theorem 2.4 we have aji=aj+(i)=ajai=aiai=a0=e. That is, an=e for the positive integer n=ji. So a has finite order, which is what we wanted to prove.

Problem 4.15 For each choice of G and each given aG list all the elements of the subgroup a of G.

  1. G=S3, a=(1 2).
  2. G=S3, a=(1 2 3).
  3. G=S4, a=(1 2 3 4).
  4. G=S4, a=(1 2)(3 4).
  5. G=Z, a=5.
  6. G=Z, a=1.
  7. G=Z15, a=5.
  8. G=Z15, a=1.
  9. G=GL(2,Z2), a=(1101).
  10. G=GL(2,R), a=(0110).

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.]


    This page titled 4: Subgroups is shared under a not declared license and was authored, remixed, and/or curated by W. Edwin Clark via source content that was edited to the style and standards of the LibreTexts platform.

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