I have to use the following theorems to determine whether or not it is possible for the given orders to be simple.
Theorem 1: |G|=1 or prime, then it is simple.
Theorem 2: If |G| = (2 times an odd integer), the G is not simple.
Theorem 3: n is an element of positive integers, n is not prime, p is prime, and p|n.
If 1 is the only divisor of n that is congruent to 1 (mod p) then no group of order n is simple.
Theorem 4: If n = p ^ k , where p is prime, then |G| = n implies G is not simple.
Theorem 5: n = ( p sub 1)(p sub 2), where (p sub 1) and (p sub 2) are both greater than 2 -- primes not simple.
I'm also going to include attachments of the book's verbatim on these theorems for clarity after the picture of the problem. I'm understanding the first two theorems, I'm not understanding how to use 3-5.
Homework Answers
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I have to use the following theorems to determine whether or not
it is possible for...
Please provide the theorems and definitions you use. 1. Let K be a subgroup of a...
Please provide the theorems and definitions you use.
1. Let K be a subgroup of a group G. Let T denote the set of all distinct right cosets of K in G and A(T) be the permutation group of T. Prove the following statements. (a) For each a EG, the function fa:T T given by fa(Kb) = Kba is a bijection. (b) The function : G + A(T) given by pla) = fa-1 is a group homomorphism whose kernel is...
The Sylow theorems state the following facts about a finite group G, of order |G| =...
The Sylow theorems state the following facts about a finite group G, of order |G| = p^m (with p prime, k positive integer, and p not dividing m) a Sy1: There exist subgroups in G of size p*, called Sylow p-subgroups particular prime p, are conjugate Sy2: All Sylow p-subgroups in G, for a Sy3: The number of Sylow p-subgroups in G is congruent to 1 modulo p, and this number divides m Consider the symmetric group S9 of permutations...
Let a : G + H be a homomorphism. Which of the following statements must necessarily...
Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...
1. Show that the set of rational numbers of the form m /n, where m, n...
1. Show that the set of rational numbers of the form m /n, where m, n E Z and n is odd is a subgroup of QQ under addition. 2. Let H, K be subgroups of a group G. Prove: H n K is a subgroup of G 3. Let G be an abelian group. Let S-aEG o(a) is finite . Show that S is a subgroup of G 4. What is the largest order of a permutation in S10?...
please look at red line please explain why P is normal thanks Proposition 6.4. There are (up to isomorphism) exactly three di groups of order 12: the dihedral group De, the alternating group A,...
please look at red line
please explain why P is normal
thanks
Proposition 6.4. There are (up to isomorphism) exactly three di groups of order 12: the dihedral group De, the alternating group A, and a generated by elements a,b such that lal 6, b a', and ba a-b. stinct nonabelian SKETCH OF PROOF. Verify that there is a group T of order 12 as stated (Exercise 5) and that no two of Di,A,T are isomorphic (Exercise 6). If G...
Can you explain these for me..I mean give example for each of them to get the...
Can you explain these for me..I mean give example for
each of them to get the idea
THE FIRST ISOMORPHISM THEOREM Let: G M be a homomorphism with Ker() = K, and Im() = I. Then there is a natural isomorphism 0:1 ™G/K which is surjective. Thus, I G/K. THE SECOND ISOMORPHISM THEOREM Suppose the N is a normal subgroup of G, and that H is a subgroup of G. Then H/( HN) = (HN)/N. THE THIRD ISOMORPHISM THEOREM Let...
Let G be a group of order 231 = 3 · 7 · 11. Let H,...
Let G be a group of order 231 = 3 · 7 · 11. Let H, K and N
denote sylow 3,7 and 11-subgroups of G, respectively.
a) Prove that K, N are both proper subsets of G.
b) Prove that G = HKN.
c) Prove that N ≤ Z(G). (you may find below problem useful).
a): <|/ is a normal subgroup, i.e. K,N are normal subgroups
of G
(below problem): Let G be a group, with H ≤ G...
Please prove C D E F in details? 'C. Let G be a group that is...
Please prove C D E F in details?
'C. Let G be a group that is DOE smDe Follow the steps indicated below; make sure to justify all an Assuming that G is simple (hence it has no proper normal subgroups), proceed as fo of order 90, The purpose of this exercise is to show, by way of contradiction. How many Sylow 3sukgroups does G have? How many Sylow 5-subgroups does G ht lain why the intersection of any two...
1-5 theorem, state it. Define all terms, e.g., a cyclic group is generated by a single...
1-5
theorem, state it. Define all terms, e.g., a cyclic group is generated by a single use a element. T encourage you to work together. If you find any errors, correct them and work the problem 1. Let G be the group of nonzero complex numbers under multiplication and let H-(x e G 1. (Recall that la + bil-b.) Give a geometric description of the cosets of H. Suppose K is a proper subgroup of H is a proper subgroup...
I help help with 34-40 33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(m...
I help help with 34-40
33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(min-1n-)1) 35. Prove that (12) and (123 m) generate S 36. Prove Cayley's theorem, which is the followving: Any finite group is isomorphic to a subgroup of some S 37. Let Dn be the dihedral group of...
Please provide the theorems and definitions you use.
1. Let K be a subgroup of a group G. Let T denote the set of all distinct right cosets of K in G and A(T) be the permutation group of T. Prove the following statements. (a) For each a EG, the function fa:T T given by fa(Kb) = Kba is a bijection. (b) The function : G + A(T) given by pla) = fa-1 is a group homomorphism whose kernel is...
The Sylow theorems state the following facts about a finite group G, of order |G| = p^m (with p prime, k positive integer, and p not dividing m) a Sy1: There exist subgroups in G of size p*, called Sylow p-subgroups particular prime p, are conjugate Sy2: All Sylow p-subgroups in G, for a Sy3: The number of Sylow p-subgroups in G is congruent to 1 modulo p, and this number divides m Consider the symmetric group S9 of permutations...
Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...
1. Show that the set of rational numbers of the form m /n, where m, n E Z and n is odd is a subgroup of QQ under addition. 2. Let H, K be subgroups of a group G. Prove: H n K is a subgroup of G 3. Let G be an abelian group. Let S-aEG o(a) is finite . Show that S is a subgroup of G 4. What is the largest order of a permutation in S10?...
please look at red line
please explain why P is normal
thanks
Proposition 6.4. There are (up to isomorphism) exactly three di groups of order 12: the dihedral group De, the alternating group A, and a generated by elements a,b such that lal 6, b a', and ba a-b. stinct nonabelian SKETCH OF PROOF. Verify that there is a group T of order 12 as stated (Exercise 5) and that no two of Di,A,T are isomorphic (Exercise 6). If G...
Can you explain these for me..I mean give example for
each of them to get the idea
THE FIRST ISOMORPHISM THEOREM Let: G M be a homomorphism with Ker() = K, and Im() = I. Then there is a natural isomorphism 0:1 ™G/K which is surjective. Thus, I G/K. THE SECOND ISOMORPHISM THEOREM Suppose the N is a normal subgroup of G, and that H is a subgroup of G. Then H/( HN) = (HN)/N. THE THIRD ISOMORPHISM THEOREM Let...
Let G be a group of order 231 = 3 · 7 · 11. Let H, K and N
denote sylow 3,7 and 11-subgroups of G, respectively.
a) Prove that K, N are both proper subsets of G.
b) Prove that G = HKN.
c) Prove that N ≤ Z(G). (you may find below problem useful).
a): <|/ is a normal subgroup, i.e. K,N are normal subgroups
of G
(below problem): Let G be a group, with H ≤ G...
Please prove C D E F in details?
'C. Let G be a group that is DOE smDe Follow the steps indicated below; make sure to justify all an Assuming that G is simple (hence it has no proper normal subgroups), proceed as fo of order 90, The purpose of this exercise is to show, by way of contradiction. How many Sylow 3sukgroups does G have? How many Sylow 5-subgroups does G ht lain why the intersection of any two...
1-5
theorem, state it. Define all terms, e.g., a cyclic group is generated by a single use a element. T encourage you to work together. If you find any errors, correct them and work the problem 1. Let G be the group of nonzero complex numbers under multiplication and let H-(x e G 1. (Recall that la + bil-b.) Give a geometric description of the cosets of H. Suppose K is a proper subgroup of H is a proper subgroup...
I help help with 34-40
33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(min-1n-)1) 35. Prove that (12) and (123 m) generate S 36. Prove Cayley's theorem, which is the followving: Any finite group is isomorphic to a subgroup of some S 37. Let Dn be the dihedral group of...
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