If H is normal, then xHxHKCH.Since o(K) = o(H), then we obtain H = K, this shows that H is unique.Sylow's Third Theorem. The number of Sylow p-subgroups in G, for a given primep, is of the form 1+kp, where k is some non-negative integer and (1 + kp) | o(G).Proof. Let M be the set of all Sylow p-subgroup of G and H be any fixed number of

Suppose H be the unique sylow P-subgroup of G.For all g∈G,g-1Hg is a sylow p-subgroup and by…

Answered: If H is normal, then xHxH KCH. Since…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 5TFE: True or False Label each of the following statements as either true or false. If a group G contains...

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22. If and are both normal subgroups of , prove that is a normal subgroup of .
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,,,, }.
9. Find all homomorphic images of the octic group.
Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
Show that An has index 2 in Sn, and thereby conclude that An is always a normal subgroup of Sn.
Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
If H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G.
13. Assume that are subgroups of the abelian group . Prove that if and only if is generated by
18. If is a subgroup of , and is a normal subgroup of , prove that .
Let G1 and G2 be groups with respect to addition. Define equality and addition in the Cartesian product by G1G2 (a,b)=(a,b) if and only if a=a and b=a (a,b)+(c,d)=(ac,bd) Where indicates the addition in G1 and indicates the addition in G2. Prove that G1G2 is a group with respect to addition. Prove that G1G2 is abelian if both G1 and G2 are abelian. For notational simplicity, write (a,b)+(c,d)=(a+c,b+d) As long as it is understood that the additions in G1 and G2 may not be the same binary operations. (Sec. 3.4,27, Sec. 3.5,14,15,27,28, Sec. 3.6,12, Sec. 5.1,51) Sec. 3.4,27 Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 3.5,14,15,27,28, Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an automorphism. Equality and addition are defined on in Exercise 52 of section 3.1. a. (x,y)=(y,x) b. (x,y)=(x,y) Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an isomorphism. a. (x,y)=x b. (x,y)=x+y Consider the additive groups 2, 3, and 6. Prove that 6 is isomorphic to 23. Let G1, G2, H1, and H2 be groups with respect to addition. If G1 is isomorphic to H1 and G2 is isomorphic to H2, prove that G1G2 is isomorphic to H1H2. Sec. 3.6,12 Consider the additive group of real numbers. Let be a mapping from to , where equality and addition are defined in Exercise 52 of Section 3.1. Prove or disprove that each of the following mappings is a homomorphism. If is a homomorphism, find ker , and decide whether is an epimorphism or a monomorphism. a. (x,y)=xy b. (x,y)=2x Sec. 5.1,51 Let R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). a. Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. b. Prove that RS is commutative if both R and S are commutative. c. Prove that RS has a unity element if both R and S have unity elements. d. Give an example of rings R and S such that RS does not have a unity element.
Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
Find all subgroups of the octic group D4.

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