2.5. COSETS AND LAGRANGE’S THEOREM
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2.4.17. Fix an element g in a group G. Show that the map cg W G !
G defined by cg .a/ D gag 1 is an automorphism of G. (This type of automorphism is called an inner automorphism.)
2.4.18. Show that conjugate elements of Sn have the same parity. More generally, if W Sn ! Sn is an automorphism, then preserves parity.
2.4.19.
(a) Show that the set of matrices with positive determinant is a normal subgroup of GL.n; R/.
(b) Show that D det ıT , where T is the homomorphism defined in Exercise 2.4.13 and is the sign homomorphism. Hint: Determine the range of det ıT and use the uniqueness of the sign homomorphism from Exercise 2.4.15.
2.4.20.
(a) For A 2 GL.n; R/ and b 2 Rn, define the transformation TA;b W Rn ! Rn by TA;b.x/ D Ax C b. Show that the set of all such transformations forms a group G.
(b) Consider the set of matrices A b ;
0
1
where A 2 GL.n; R/ and b 2 Rn, and where the 0 denotes a 1by-n row of zeros. Show that this is a subgroup of GL.n C 1; R/, and that it is isomorphic to the group described in part (a).
(c) Show that the map TA;b 7! A is a homomorphism from G to GL.n; R/, and that the kernel K of this homomorphism is isomorphic to Rn, considered as an abelian group under vector addition.
2.4.21. Let G be an abelian group. For any integer n > 0 show that the map ' W a 7! an is a homomorphism from G into G. Characterize the kernel of '. Show that if n is relatively prime to the order of G, then ' is an isomorphism; hence for each element g 2 G there is a unique a 2 G such that g D an.
2.5. Cosets and Lagrange’s Theorem Consider the subgroup H D fe; .1 2/g S3. For each of the six elements 2 S3 you can compute the set H D f W 2 H g. For example, .23/H D f.23/; .132/g. Do the computation now, and check that you get
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2. BASIC THEORY OF GROUPS the following results: eH D .1 2/H D H .2 3/H D .1 3 2/H D f.2 3/; .1 3 2/g .1 3/H D .1 2 3/H D f.1 3/; .1 2 3/g: As varies through S3, only three different sets H are obtained, each occurring twice.
Definition 2.5.1. Let H be subgroup of a group G. A subset of the form gH , where g 2 G, is called a left coset of H in G. A subset of the form H g, where g 2 G, is called a right coset of H in G.
Example 2.5.2. S3 may be identified with a subgroup of S4 consisting of permutations that leave 4 fixed and permute f1; 2; 3g. For each of the 24 elements 2 S4 you can compute the set S3.
This computation requires a little labor. If you want, you can get a computer to do some of the repetitive work; for example, programs for computations in the symmetric group are distributed with the symbolic mathematics program Mathematica.
With the notation H D f 2 S4 W .4/ D 4g, the results are eH D .1 2/H D .1 3/H D .2 3/H D .1 2 3/H D .1 3 2/H D H .4 3/H D .4 3 2/H D .2 1/.4 3/H D .2 4 3 1/H D .4 3 2 1/H D .4 3 1/H D f.4 3/; .4 3 2/; .2 1/.4 3/; .2 4 3 1/; .4 3 2 1/; .4 3 1/g .4 2/H D .3 4 2/H D .4 2 1/H D .4 2 3 1/H D .3 4 2 1/H D .3 1/.4 2/H D f.4 2/; .3 4 2/; .4 2 1/; .4 2 3 1/; .3 4 2 1/; .3 1/.4 2/g .4 1/H D .4 1/ .3 2/H D .2 4 1/H D .2 3 4 1/H D .3 2 4 1/H D .3 4 1/H D f.4 1/; .4 1/ .3 2/; .2 4 1/; .2 3 4 1/; .3 2 4 1/; .3 4 1/g:
The regularity of the preceding data for left cosets of subgroups of symmetric groups is striking! Based on these data, can you make any conjectures (guesses!) about properties of cosets of a subgroup H in a group G?
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Properties of Cosets Proposition 2.5.3. Let H be a subgroup of a group G, and let a and b be elements of G. The following conditions are equivalent: (a) a 2 bH .
(b) b 2 aH .
(c) aH D bH .
(d) b 1a 2 H .
(e) a 1b 2 H .
Proof. If condition (a) is satisfied, then there is an element h 2 H such that a D bh; but then b D ah 1 2 aH . Thus, (a) implies (b), and similarly (b) implies (a). Now suppose that (a) holds and choose h 2 H such that a D bh. Then for all h1 2 H , ah1 D bhh1 2 bH ; thus aH bH . Similarly, (b) implies that bH aH . Since (a) is equivalent to (b), each implies (c). Because a 2 aH and b 2 bH , (c) implies (a) and (b). Finally, (d) and (e) are equivalent by taking inverses, and a D bh 2 bH , b 1a D h 2 H , so (a) and (d) are equivalent.
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Proposition 2.5.4. Let H be a subgroup of a group G.
(a) Let a and b be elements of G. Either aH D bH or aH \ bH D ;.
(b) Each left coset aH is nonempty and the union of left cosets is G.
Proof. If aH \ bH 6D ;, let c 2 aH \ bH . By the previous proposition cH D aH and cH D bH , so aH D bH . For each a 2 G, a 2 aH ; this implies both assertions of part (b).
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Proposition 2.5.5. Let H be a subgroup of a group G and let a and b be elements of G. Then x 7! ba 1x is a bijection between aH and bH .
Proof. The map x 7! ba 1x is a bijection of G (with inverse y 7!
ab 1y). Its restriction to aH is a bijection of aH onto bH .
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Theorem 2.5.6. (Lagrange’s theorem). Let G be a finite group and H a subgroup. Then the cardinality of H divides the cardinality of G, and the jGj
quotient
is the number of left cosets of H in G.
jH j i
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2. BASIC THEORY OF GROUPS Proof. The distinct left cosets of H are mutually disjoint by Proposition 2.5.4 and each has the same size (namely jH j D jeH j) by Proposition 2.5.5. Since the union of the left cosets is G, the cardinality of G is the cardinality of H times the number of distinct left cosets of H .
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Definition 2.5.7. For a subgroup H of a group G, the index of H in G is the number of left cosets of H in G. The index is denoted ŒG W H .
Index also makes sense for infinite groups. For example, take the larger group to be Z and the subgroup to be nZ.
Then ŒZ W nZ D n, because there are n cosets of nZ in Z. Every subgroup of Z (other than the zero subgroup f0g) has the form nZ for some n, so every nonzero subgroup has finite index. The zero subgroup has infinite index.
It is also possible for a nontrivial subgroup of an infinite group (i.e., a subgroup that is neither feg nor the entire group) to have infinite index.
For example, the cosets of Z in R are in one to one correspondence with the elements of the half-open interval Œ0; 1/; there are uncountably many cosets! See Exercise 2.5.11.
Corollary 2.5.8. Let p be a prime number and suppose G is a group of order p. Then (a) G has no subgroups other than G and feg.
(b) G is cyclic, and in fact, for any nonidentity element a 2 G, G D hai.
(c) Every homomorphism from G into another group is either trivial (i.e., every element of G is sent to the identity) or injective.
Proof. The first assertion follows immediately from Lagrange’s theorem, since the size of a subgroup can only be p or 1. If a ¤ e, then the subgroup hai is not feg, so must be G. The last assertion also follows from the first, since the kernel of a homomorphism is a subgroup.
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Any two groups of prime order p are isomorphic, since each is cyclic.
This generalizes (substantially) the results that we obtained before on the uniqueness of the groups of orders 2, 3, and 5.
Corollary 2.5.9. Let G be any finite group, and let a 2 G. Then the order o.a/ divides the order of G.
Proof. The order of a is the cardinality of the subgroup hai.
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The index of subgroups satisfies a multiplicativity property:
Proposition 2.5.10. Suppose K H G are subgroups. Then ŒG W K D ŒG W H ŒH W K: Proof. If the groups are finite, then by Lagrange’s theorem,
jGj
jGj jH j ŒG W K D
D D ŒG W H ŒH W K:
jKj
jH j jKj
If the groups are infinite, we have to use another approach, which is discussed in the Exercises.
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Definition 2.5.11. For any group G, the center Z.G/ of G is the set of elements that commute with all elements of G, Z.G/ D fa 2 G W ag D ga for all g 2 Gg:
You are asked in the Exercises to show that the center of a group is a normal subgroup, and to compute the center of several particular groups.
Exercises 2.5 2.5.1. Check that the left cosets of the subgroup K D fe; .123/; .132/g in S3 are eK D .123/K D .132/K D K .12/K D .13/K D .23/K D f.12/; .13/; .23/g and that each occurs three times in the list .gK/g2S . Note that K is the
3
subgroup of even permutations and the other coset of K is the set of odd permutations.
2.5.2. Suppose K H G are subgroups. Suppose h1K; : : : ; hRK is a list of the distinct cosets of K in H , and g1H; : : : ; gS H is a list of the distinct cosets of H in G. Show that fgshr H W 1 s S; 1 r Rg is the set of distinct cosets of H in G. Hint: There are two things to show.
First, you have to show that if .r; s/ ¤ .r0; s0/, then gshr K ¤ gs0hr0K.
Second, you have to show that if g 2 G, then for some .r; s/, gK D gshr K.
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2. BASIC THEORY OF GROUPS 2.5.3. Try to extend the idea of the previous exercise to the case where at least one of the pairs K H and H G has infinite index.
2.5.4. Consider the group S3.
(a) Find all the left cosets and all the right cosets of the subgroup H D fe; .12/g of S3, and observe that not every left coset is also a right coset.
(b) Find all the left cosets and all the right cosets of the subgroup K D fe; .123/; .132/g of S3, and observe that every left coset is also a right coset.
2.5.5. What is the analogue of Proposition 2.5.3, with left cosets replaced with right cosets?
2.5.6. Let H be a subgroup of a group G. Show that aH 7! Ha 1 defines a bijection between left cosets of H in G and right cosets of H in G. (The index of a subgroup was defined in terms of left cosets, but this observation shows that we get the same notion using right cosets instead.)
2.5.7. For a subgroup N of a group G, prove that the following are equivalent: (a) N is normal.
(b) Each left coset of N is also a right coset. That is, for each a 2 G, there is a b 2 G such that aN D N b.
(c) For each a 2 G, aN D Na.
2.5.8. Suppose N is a subgroup of a group G and ŒG W N D 2. Show that N is normal using the criterion of the previous exercise.
2.5.9. Show that if G is a finite group and N is a subgroup of index 2, then for elements a and b of G, the product ab is an element of N if, and only if, either both of a and b are in N or neither of a and b is in N .
2.5.10. For two subgroups H and K of a group G and an element a 2 G, the double coset HaK is the set of all products hak, where h 2 H and k 2 K. Show that two double cosets HaK and H bK are either equal or disjoint.
2.5.11. Consider the additive group R and its subgroup Z. Describe a coset t C Z geometrically. Show that the set of all cosets of Z in R is ft C Z W 0 t