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# 代考数学考试 MT4003代写 数学考试代考 数学考试代写

78

## MT4003 Groups

EXAM DURATION: 2 hours

EXAM INSTRUCTIONS: Attempt ALL questions.

The number in square brackets shows the maximum marks obtainable for that question or part-question.

INSTRUCTIONS FOR ONLINE EXAMS:

Each page of your solution must have the page number, module code, and your student ID number at the top of the page. You must make sure all pages of your solutions are clearly legible.

### 1.代考数学考试

Let G be the subgroup of S9 generated by the permutations

α = (1 3)(2 7), β = (1 7)(2 5)(3 9).

Throughout, you may use results from lectures, provided that they are clearly stated.

(a) What are the orders of αβ and βα? 

(b) Let H be the cyclic subgroup of G generated by the permutation (αβ)2 . Determine |H|, and show that H is not a normal subgroup of S9. 

(c) By considering the number of elements of G calculated so far, as well as the orders of these elements, prove that |G| ≥ 12. 

(You do not have to find all of the elements of G). 代考数学考试

(d) Show that for any h H, the conjugates α1and β1are also elements of H. Briefly explain why this is sufficient to show that H is a normal subgroup of G. 

For the rest of this question, you are given that |G| = 12.

(e) Let K be a subgroup of G, such that

H K G,

where H is the subgroup from Part (b). Suppose that |K| ≠ |G|, and |K| ≠ |H|. Determine the order of K, and give an example of a subgroup K matching all the above criteria. 

(f) Deduce that H is a normal subgroup of K, and that K is a normal subgroup of G. Use the second isomorphism theorem to prove that the quotient group KH/K is isomorphic to the trivial group. 

### 2.代考数学考试

Give an example of each of the following, fully justifying your answers. You may use results from lectures, provided that they are clearly stated.

(a) Two non-isomorphic groups of order 90. 

(b) A group with a nontrivial normal subgroup of index 3. 

(c) A group with exactly two elements of order 3. 

(d) A non-abelian group with exactly eight elements of order 3.

(e) A non-abelian group of order 48 that is not isomorphic to a dihedral group. 

(f) A non-abelian group whose derived subgroup has index 5. 

### 3.代考数学考试

Recall that a finite p-group is a group whose order is a power of a prime p. Let G be a p-group of order pn with n 1.

Throughout, you may use results from lectures, provided that they are clearly stated.

(a) Prove that if H G and x is an element of G that does not lie in H then |x, H : H| ≥ p. Hence or otherwise show that if |G| = pn then G can be generated by n elements. 

(b) Give an example of a group of order 16 that requires 4 elements to generate it, fully justifying your answer. 

(c) Prove that every homomorphic image of a p-group is a p-group. 

(d) For any group K, recall that Z(K) denotes the centre of K. Define an infinite sequence of groups by K(0) = K, K(1) = K(0)/Z(K(0)), . . . , K(i) = K(i−1)/Z(K(i−1)), . . .. 代考数学考试

(i) Prove by induction on n that |G(n)| = 1, where G is a group of order pn as before. 

(ii) Determine the smallest value of i such that |G(i)| = 1 when G is each of the following groups: the cyclic group of order 8; the quaternion group Q8 of order 8. 

(iii) Give an example of a group K (which will not be a p-group) such that K(i) 6 = 1 for all i. Fully justify your answer. 

### 4.

Throughout, you may use results from lectures, provided that they are clearly stated.

(a) For each of the following group orders, determine whether there is a simple group of that order: (i) 126; (ii) 360; (iii) 520. 

The prev:

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