Practise Questions:
The octahedron is the regular solid which is formed by stacking two
square based pyramids base to base.
It has 6 vertices
and 8 faces, how many edges are there?
List the different types of rotations through vertices, edges
and faces, and explain why there are 24 in total.
Using Cauchy-Frobenius, calculate the number of ways to
colour 2 or 3 vertices black
Deduce the numbers of ways to
colour 0, 1, 4, 5 or 6 and draw pictures of them all.
How many different black colourings of 2 faces and 1 edge are there?
What about if you use n colours for the edges?
What are all of the different homomorphisms between
D4 and Z4?
How about D5 and Z5? Dp and Zp for p prime?
Z2 x Z2 acts by symmetry as a group action on the set of areas:
Write out the group action table,
find the orbits of the action, and identify the stabilizers of p, q and r.
Verify the orbit-stabiliser relation for each orbit.
Identify the fixed elements and
verify that Cauchy Frobenius holds in this case.
How many shadings of the pattern are there with 4 areas coloured?
Midterm Practise Questions:
Prove that if H is a normal subgroup of G1 and K is an ordinary subgroup of G2 then:
-
H x K is a subgroup of G1 x G2
{e} x K is a subgroup of G1 x G2
H x {e} is a normal subgroup of G1 x G2
Prove that G1 x G2 is Abelian if and only if both G1 and G2 are
Deduce that H x K is a normal subgroup of G1 x G2 if G2 is Abelian
List all the elements in Z2 x Z6 and Z3 x Z4 and their orders and hence explain why the groups are non-isomorphic.
Which of these group do
other products of numbers which multiply to 12 gives rise to?
Choose appropriate
elements from each group and complete the Cayley diagram
for each group based upon these. What rules do the two elements follow?
By finding elements of specific orders in A4, D3 x Z2 and T show that
all of these groups are non-isomorphic.
For H and K subgroups of G, give examples of when H union K is
and is not a subgroup of G.
If H is not a subgroup of K (or vice versa) can H union K ever be a subgroup of G?
If H is not a subgroup of G can H union K ever be a subgroup of G?
What is the order of H union K?
What is H, the cyclic subgroup of S6 generated by (163)(45) ?
Is it a normal subgroup of S6?
What is the cyclic subgroup K of S6 generated by (163)(245) ?
Find an element not in K which is in the centraliser of (163)(245)
Are there any such elements in H?
Show that there exists a normal subgroup N isomorphic to Z2 x Z2 in A4
List the elements of A4/N and verify that |A4/V| = |A4| / |V| if V is the Klein 4-group which is Z2 x Z2
Prove that if G is Abelian then G/H will be too, but the converse
does not have to be true
Prove that |G/N| = |G| / |N| for any normal subgroup N.
Programs:
PermGroup @wims.unice.fr (webGAP)
Groups and Graphs (for Windows)