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Chapter 1:
Groups
Determine the order of your group.
Find the inverses of the first 10 elements.
Write your answer as a list with square brackets:
[ inverse of A, inverse of B, ... , inverse of J]
Determine the orders of the first 10 elements.
Write your answer as a list with square brackets:
[ order of A, order of B, ... , orde of J]
Chapter 2:
Subgroups, homomorphisms, direct products
Give 3 different non-trivial subgroups.
Write your answer as a list with square brackets: e.g. [A,B,E,F].
Subgroup 1:
Subgroup 2:
Subgroup 3:
Chapter 3:
Generators, order, index
Find a list of generators for your group.
Write your answer as a list with square brackets:
Give a list with at least 3 relations between your generators.
(A relation is an expression in the generators that evaluates to the unit element e.g. B^3*C^2)
Choose a non-trivial subgroup and determine the left and right cosets.
Make sure that the left and right cosets are different.
Write your answer as 3 lists with square brackets: e.g.
[A,B,D,L]
[[A,B,D,L],[E,F,G,H],[I,J,K,C]]
[[A,B,D,L],[I,J,G,H],[F,E,K,C]]
Subgroup:
Right cosets
Left cosets
What is the index of this subgroup?
Chapter 4:
Normal subgoups, Quotients
Give 3 different non-trivial normal subgroups.
Write your answer as a list with square brackets e.g. [A,B,E,F].
Determine to which known group the quotient is isomorphic.
Choose normal subgroups for which the quotient is in the list.
If you think there are less than 3 normal subgroups, fill in NONE for the third.
Normal subgroup 1:
Quotient 1:
");
C2
C3
C4
C2 x C2
C5
C6
S3
C8
C2 x C2 x C2
D8
D10
C12
D12
A4
C14
D14
C16
D16
C18
D18
(C3 x C3) ⋊ C2
C20
D20
C22
D22
C24
D24
C25
C5 x C5
C26
D26
Normal subgroup 2:
Quotient 2:
");
C2
C3
C4
C2 x C2
C5
C6
S3
C8
C2 x C2 x C2
D8
D10
C12
D12
A4
C14
D14
C16
D16
C18
D18
(C3 x C3) ⋊ C2
C20
D20
C22
D22
C24
D24
C25
C5 x C5
C26
D26
Normal subgroup 3:
Quotient 3:
");
C2
C3
C4
C2 x C2
C5
C6
S3
C8
C2 x C2 x C2
D8
D10
C12
D12
A4
C14
D14
C16
D16
C18
D18
(C3 x C3) ⋊ C2
C20
D20
C22
D22
C24
D24
C25
C5 x C5
C26
D26
Determine the center of your group and write it as a list:
Determine the commutator subgroup of your group and write it as a list:
Chapter 5:
Homomorphism and isomorphism theorems
Find a someone with a
group
for which there is a non-trivial morphism f from your group to his/her group.
Name of the person:
Group Id of the person:
Write the images of your generators in a list
[f(1st generator), f(2nd generator), ...]
Determine the kernel of f as a list.
Determine the image of f as a list.
Chapter 6:
Group Action
Find an as small as possible set on which your group acts non-trivially.
Number the elements of the set as 1,2,3,... and determine the permutations
of the generators in cycle notation. The trivial permutation is written as ().
Put kommas between the numbers in the cycles.
E.g. (1,2)(3,4) swaps 1 and 2, and 3 and 4.
Write your answer as a list. e.g. [(1,2)(3,4),(3,4,1),()]
You get an extra mark if your action faithful (i.e. the map G -> Sn is injective)
Hint: Look at conjugation or left mutiplication on cosets of a subgroup.
Determine the stabilizer of the 2nd element in the set. Write it as a list.
Determine the orbit of the 2nd element.
Write it as a list of numbers e.g. [2,4,5].
How many orbits are there?
Determine the conjugacy classes of your group
Write your answer as a list of lists:
[ [ A ], [ E, B,D ], [C,F], ...]
Chapter 7:
Automorphisms
Find a non-trivial internal automorphism of your group. Write the images of your generators in a list
[f(1st generator), f(2nd generator), ...]
How many internal automorphisms are there?
Find a non-trivial external automorphism of your group. Write the images of your generators in a list
[f(1st generator), f(2nd generator), ...]
Fill in NONE if there is none
How automorphisms are there?
Find a normal subgroup N and a subgroup K such that your group is the semidirect product of N and K.
Give your answer as 3 list:
A list of generators for N
A list of generators for K
The group action of K on N. Write this as a list of lists. Th nth list contains the images of the generators of N under the nth generator of K
Chapter 8:
Finite abelian groups
Below is a list of all abelian groups with order below 27
C2, C3, C4, C2xC2, C5, C6, C7, C8, C4xC2, C2xC2xC2, C9, C3xC3, C10, C11, C12, C6xC2, C13, C14, C15, C16, C4xC4, C8xC2, C4xC2xC2, C2xC2xC2xC2, C17, C18, C6xC3, C19, C20, C10xC2, C21, C22, C23, C24, C12xC2, C6xC2xC2, C25, C5xC5, C26
Determine for each of these whether there is a subgroup of your group isomorphic to it.
Give a list of all groups that occur. e.g. [C2,C3,C2xC2]
Take care: use the names above (so not C5xC2 but C10).
Give for each type (a list of generators of) a subgroup of that type. e.g. [[A,B],[A,C,D],[A,E,F,G]]
Give for each type the number of subgroups of that type. e.g. [3,5,1]
What is your group?
Select the correct name for your group from the list.
(C3 x C3) ⋊ C2
C5 ⋊ C4
C5 ⋊ C4
C3 ⋊ C8
C3 ⋊ Q8
C2 x (C3 ⋊ C4)
(C6 x C2) ⋊ C2
C7 ⋊ C4
C9 ⋊ C4
(C2 x C2) ⋊ C9
C3 x (C3 ⋊ C4)
(C3 x C3) ⋊ C4
(C3 x C3) ⋊ C4
C2 x ((C3 x C3) ⋊ C2)
C13 ⋊ C3
C5 ⋊ C8
C5 ⋊ C8
C5 ⋊ Q8
C2 x (C5 ⋊ C4)
(C10 x C2) ⋊ C2
C2 x (C5 ⋊ C4)
(C7 ⋊ C3) ⋊ C2
C2 x (C7 ⋊ C3)
C11 ⋊ C4
C3 ⋊ C16
(C4 x C4) ⋊ C3
C24 ⋊ C2
C24 ⋊ C2
C3 ⋊ Q16
C2 x (C3 ⋊ C8)
(C3 ⋊ C8) ⋊ C2
C4 x (C3 ⋊ C4)
(C3 ⋊ C4) ⋊ C4
C12 ⋊ C4
(C12 x C2) ⋊ C2
(C3 x D8) ⋊ C2
(C3 ⋊ C8) ⋊ C2
(C3 x Q8) ⋊ C2
C3 ⋊ Q16
(C2 x (C3 ⋊ C4)) ⋊ C2
C3 x ((C4 x C2) ⋊ C2)
C3 x (C4 ⋊ C4)
C3 x (C8 ⋊ C2)
A4 ⋊ C4
SL(2,3) ⋊ C2
C2 x (C3 ⋊ Q8)
(C12 x C2) ⋊ C2
(C2 x (C3 ⋊ C4)) ⋊ C2
(C4 x S3) ⋊ C2
C2 x C2 x (C3 ⋊ C4)
C2 x ((C6 x C2) ⋊ C2)
C3 x ((C4 x C2) ⋊ C2)
(C2 x C2 x C2 x C2) ⋊ C3
(C5 x C5) ⋊ C2
C13 ⋊ C4
C13 ⋊ C4
Honours extension (Score: /5)
Find a decomposition series for your group. If
G
_{0}
=[A] < G
_{1}
< ... < G
_{k}
=G
you fill in your answer as a list with generators for the groups G
_{0}
to G
_{k}
.
e.g. [[A],[B],[B,D],[B,D,E]]
Give a list with descriptions of the decomposition factors G
_{i}
/G
_{i-1}
in the same order as your decomposition series.
e.g. [C2,C2,C3].
Give a list with all orders of Sylow subgroups.
e.g [4,3,25]
For each order give the generators of one Sylow subgroup with that order.
e.g. [[B,C],[E],[G,H]]
For each order determine the number of Sylow subgroups with that order
e.g. [3,1,6]