Take Home test Group Theory
Ministério da Ciência, 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 LATEX will also be evaluated.
Questions
1. Let V be a finite dimension vector space over a field F . Let α ∈ GL(V ) and a ∈ V . A
function fα,a : V → V defined by xfα,a = xα + a is said to be an affine transformation. A
map fι,a , where ι 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 of G.
3. Denote by H the group of translations. Prove that G = H · GL(V ).
2. A near field is an algebra (F, +, ·) in which (F, +) and (F, ·) are semigroups, (F, +) is
a group with identity 0 and (F \ {0}, ·) is a group; the two operations are connected by
right distributivity
(x + y) · z = x · z + y · z.
Let F be a finite near-field. For all a ∈ F \ {0} and b ∈ F define fa,b : F → F by
xfa,b = xa + b, for all x ∈ F . Let G be the set of all fa,b with the usual composition of
transformations.
1. Prove that G is a group.
2. Prove that G is 2-transitive.
3. A group H of permutations of Ω is said to be sharply 2-transitive if given (x, y), (u, v) ∈
Ω2 , with x, y, u, v all different, there exists one and only one h ∈ H such that
(x, y)h = (u, v).
Prove that G is sharply 2-transitive.
3. Let V be a finite dimension vector space over a field F . Denote by P GL(V ) the factor
group GL(V )/Z, where Z denotes the center of GL(V ).
Let V be a 2-dimensional vector space over a field. Prove that there exists a monomorphism from P GL(V ) to a 3-transitive permutation group given by the action of
P GL(V ) on the 1-dimension subspaces of V .
4. Consider the group G generated by the following permutations of {1, . . . , 12}:
(1, 2, 3)(4, 5, 6)(7, 8, 9)
(2, 4, 3, 7)(5, 6, 9, 8)
(2, 9, 3, 5)(4, 6, 7, 8)
(1, 10)(4, 7)(5, 6)(8, 9)
(4, 8)(5, 9)(6, 7)(10, 11)
(4, 7)(5, 8)(6, 9)(11, 12)
1. Find the size of G.
1
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 that Ag = B.
Find all the k ≤ 12 such that G is k-homogeneous.
3. Find the largest k such that G is 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 order o (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?
The End
2
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 LATEX will also be evaluated.
Questions
1. Let V be a finite dimension vector space over a field F . Let α ∈ GL(V ) and a ∈ V . A
function fα,a : V → V defined by xfα,a = xα + a is said to be an affine transformation. A
map fι,a , where ι 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 of G.
3. Denote by H the group of translations. Prove that G = H · GL(V ).
2. A near field is an algebra (F, +, ·) in which (F, +) and (F, ·) are semigroups, (F, +) is
a group with identity 0 and (F \ {0}, ·) is a group; the two operations are connected by
right distributivity
(x + y) · z = x · z + y · z.
Let F be a finite near-field. For all a ∈ F \ {0} and b ∈ F define fa,b : F → F by
xfa,b = xa + b, for all x ∈ F . Let G be the set of all fa,b with the usual composition of
transformations.
1. Prove that G is a group.
2. Prove that G is 2-transitive.
3. A group H of permutations of Ω is said to be sharply 2-transitive if given (x, y), (u, v) ∈
Ω2 , with x, y, u, v all different, there exists one and only one h ∈ H such that
(x, y)h = (u, v).
Prove that G is sharply 2-transitive.
3. Let V be a finite dimension vector space over a field F . Denote by P GL(V ) the factor
group GL(V )/Z, where Z denotes the center of GL(V ).
Let V be a 2-dimensional vector space over a field. Prove that there exists a monomorphism from P GL(V ) to a 3-transitive permutation group given by the action of
P GL(V ) on the 1-dimension subspaces of V .
4. Consider the group G generated by the following permutations of {1, . . . , 12}:
(1, 2, 3)(4, 5, 6)(7, 8, 9)
(2, 4, 3, 7)(5, 6, 9, 8)
(2, 9, 3, 5)(4, 6, 7, 8)
(1, 10)(4, 7)(5, 6)(8, 9)
(4, 8)(5, 9)(6, 7)(10, 11)
(4, 7)(5, 8)(6, 9)(11, 12)
1. Find the size of G.
1
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 that Ag = B.
Find all the k ≤ 12 such that G is k-homogeneous.
3. Find the largest k such that G is 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 order o (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?
The End
2