Proposition For any s€ S, the set stab(s) is a subgroup of G. Proof (outline) To show stab(s) is a group we need to show three things: (i) Identity. That is, 5.4(1) = s. (i) Inverses. That is if s.o(g)-s, then s.(g¹)=s. (iii) Closure. That is, if s.p(g) =s and s.(n) =s, then s.(gh) = s. Alternatively, it suffices to show that if s.(g) =s and s.d(h) = s. then s.0(gh¹) = 5. The stabilizer subgroup Here is another example of an action, this time of G = De- Let's be the highlighted hexagon, and Hstab(s). Hr² Hr (121) (148) Let G be a finite group acting on a set X, and let z X be an element of the set. The stabilizer of , denoted Stabc(z), is defined as the set of all elements in G that fix 2 under the group action, The stabilizer of s in G is i.e., stab(s) {gG|s.(g) = s}. The stabilizer subgroup As we've seen, elements in the same orbit can have different stabilizers. Proposition (HW exercise) Set elements in the same orbit have conjugate stabilizers: stab(s.$(g))=gstab(s)g. for all gЄ G and s € S. In other words, if x stabilizes s, then g¹xg stabilizes s. (g). Here are several ways to visualize what this means and why. Prove the following: Stabc() {gGg-2=2} The stabilizer of z is a subgroup of G. If G acts transitively on X, then the size of the orbit of a is given by G-z]=[G: Stab()], where [G: Staba (z)] denotes the index of the stabilizer subgroup in G. Here is the action graph of the group D₁ = {r, f): 2 Hr³ H+4 labeled by destinations Hr Н labeled by stabilizers 0(x) 5 $(1) (9) (9) $(r) (r) (fr)=(1) (124) In other words, if x is a loop from s, and ss, then g¹xg is a loop from s'. In the beginning of this course, we picked a configuration to be the "solved state," and this gave us a bijection between configurations and actions (group elements).

Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.

Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.

Let G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? Exercise 3 b. Exercise 4 c. Exercise 5 d. Exercise 6 e. Exercise 7 f. Exercise 8 Section 4.4 Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup (1),(2,3) of S3. Find the distinct left cosets of H in S3, write out their elements, partition S3 into left cosets of H, and give [S3:H]. Find the distinct right cosets of H in S3, write out their elements, and partition S3 into right cosets of H. In Exercises 7 and 8, let G be the multiplicative group of permutation matrices I3,P3,P32,P1,P4,P2 in Example 6 of Section 3.5 Let H be the subgroup of G given by H=I3,P4={ (100010001),(001010100) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H. Let H be the subgroup of G given by H=I3,P3,P32={ (100010001),(010001100),(001100010) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H.

5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:

9. Find all homomorphic images of the octic group.

27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.

9. Find all elements in each of the following groups such that . under addition. under multiplication.

Find all subgroups of the octic group D4.

Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .