Minist´erio da Ciˆencia, Tecnologia e Ensino Superior
Take Home Test Group Theory 21–25 January 2016
INSTRUCTIONS
-• Solve any two questions you wish of the questions on the list below. Each question is worth 2.5 points.
• You can solve more than two, and only the best 2 answers will be considered for credit.
• This activity is individual work. Collaboration is not permitted.
Evaluation Criteria:
• Mathematical correctness of the answers is the most valued aspect of this activity.
• Clear and correct mathematical writing is the second most valued item. Please take your time to make each point clear. Your answers should not assume the reader is an expert on the topic, rather they should be thought of as a way of explaining the solution of the problem to someone that is just learning the topic.
• Proper use of LA
Questions
1. LetV be a finite dimension vector space over a fieldF. Letα∈GL(V) and a∈V. A functionfα,a :V →V defined byxfα,a =xα+ais said to be an affine transformation. A
map fι,a, whereι is the identity on V, is said to be a translation.
1. Prove that the set of affine transformations forms a group.
2. Let G be the group of affine transformations. Prove that the translations form a normal subgroup ofG.
Let V be a 2-dimensional vector space over a field. Prove that there exists a mo-nomorphism from P GL(V) to a 3-transitive permutation group given by the action of
2. A group of permutations of Ω is said to be k-homogeneous (for a k < |Ω|) if given any two k-sets A, B ⊆Ω, we have g ∈G such thatAg=B.
Find all the k ≤12 such that Gis k-homogeneous.
3. Find the largest k such that Gis k-transitive.
4. Find the name under which G is usually known.
5. Prove, without using GAP, that any group of order 15 is cyclic.
6. Let o be a natural number such that 55 ≤ o ≤ 63. Without using GAP, answer the following questions.
1. Identify the values of o for which there exists a cyclic group of order o.
2. Prove that a simple group of ordero (recall that 55≤o≤63) either is cyclic or has order 60.
7. Find, without using GAP, the orders of the groups A8, PSL(4,2), and PSL(3,4). Decide (possibly using GAP) if any pairs of these groups are isomorphic?
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