# My favourite group (Score: /40)

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## Chapter 1:Groups

Determine the order of your group.

Find the inverses of the first 10 elements.
[ inverse of A, inverse of B, ... , inverse of J]

Determine the orders of the first 10 elements.
[ order of A, order of B, ... , orde of J]

## Chapter 2:Subgroups, homomorphisms, direct products

Give 3 different non-trivial subgroups.
Subgroup 1:

Subgroup 2:

Subgroup 3:

## Chapter 3:Generators, order, index

Find a list of generators for your group.

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.
[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.
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:

Normal subgroup 2:

Quotient 2:

Normal subgroup 3:

Quotient 3:

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.

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
[ [ 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.
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]

Select the correct name for your group from the list.

# Honours extension (Score: /5)

Find a decomposition series for your group. If
G0=[A] < G1 < ... < Gk=G
you fill in your answer as a list with generators for the groups G0 to Gk.
e.g. [[A],[B],[B,D],[B,D,E]]

Give a list with descriptions of the decomposition factors Gi/Gi-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]