Week 7 Problem Sheet
Group Theory and Linear algebra
Semester II 2011

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 1 September 2011

(1) Week 7: Vocabulary
(2) Week 7: Results
(3) Week 7: Examples and computations

Week 7: Vocabulary

Define a group and give some illustrative examples.
Define abelian group and give some illustrative examples.
Define permutations and give some illustrative examples.
Define a symmetric group and give some illustrative examples.
Define a cyclic group and give some illustrative examples.
Define cyclic subgroup generated by g and give some illustrative examples.
Define GLn(R) and give some illustrative examples.
Define On(R) and give some illustrative examples.
Define Un() and give some illustrative examples.
Define SLn(R) and give some illustrative examples.
Define SOn(R) and give some illustrative examples.
Define SUn() and give some illustrative examples.
Define subgroup and give some illustrative examples.
Define subgroup generated by g1,, gk and give some illustrative examples.
Define order of a group and order of an element of a group and give some illustrative examples.
Define homomorphism and isomorphism and give some illustrative examples.
Define product of groups and give some illustrative examples.
Define kernel and image of a group homomorphism and give some illustrative examples.

Week 7: Results

Let G be a group. Show that the identity of G is unique.
Let G be a group and let g,hG. Show that (gh) -1 = h-1 g-1 .
Let G be a group and let g,x,y G. Show that if gx=gy then x=y.
Let G be a group and let g,h G. Show that there exist unique x,yG such that gx=h and yg=h.
Let G be a subgroup. Show that H is a subgroup of G if and only if H is a subset of G such that if h1,h2H then h1 h2 H and h1-1H .
Let G be a subgroup. Show that H is a subgroup of G if and only if H is a subset of G such that if h1,h2H then h1 h2-1H .
Show that every subgroup of a cyclic group is cyclic.
Show that if G is a cyclic group then G is isomorphic to or there exists n>0 such that G is isomorphic to /n.
Let f:GH be a group homomorphism. Show that f(1)=1.
Let f:GH be a group homomorphism and let gG. Show that f(g-1) =f(g) -1.
Let f:GH be a group homomorphism and let gG. Show that the order of g is equal to the order of f(g).
Let G and H be groups. Prove that G×H with operation defined by (g1,h1) (g2,h2) = (g1g2, h1h2) is a group.

Week 7: Examples and computations

Let n be a positive integer. Show that the set of all complex nth roots of unity {z | zn=1} forms a group under multiplication.
Let U(n) be the set of n×n unitary matrices. Show that U(n) is a group under matrix multiplication.
Let G be a group and let x,y,z,wG. Assume that xyz-1w =1. Solve for y.
Compute the following products of permutations: (123) (456) * (12) (34) (56) and (12)* (246)* (123654) .
Write out the multiplication table for the group S3 of permutations of {1,2,3} using cycle notation.
Let G be a group and let x,y,zG. Assume that xyz=1. Does it follow that yzx=1? Does it follow that yxz=1?
Assume that G is a group such that if gG then g2=1. Show that G is abelian.
Show that with the operation of addition is a group.
Show that with the operation of addition is a group.
Show that with the operation of addition is a group.
Show that with the operation of addition is a group.
Show that with the operation of multiplication is not a group.
Show that with the operation of multiplication is not a group.
Show that with the operation of multiplication is not a group.
Show that with the operation of multiplication is not a group.
Show that Mn() with the operation of addition is a group.
Show that GLn() is a group.
Show that On() is a group.
Show that SLn() is a group.
Show that SOn() is a group.
Describe the elements of GL1() and GL2().
Describe the elements of GL1() and GL2().
Describe the elements of SL1() and SL2().
Describe the elements of SL1() and SL2().
Describe the elements of O1() and O2().
Describe the elements of O1() and O2().
Describe the elements of SO1() and SO2().
Describe the elements of SO1() and SO2().
Describe the elements of U1(), SU1(), U2(), and SU2().
Describe the elements of On().
Describe the elements of SOn().
Find the multiplication tables of all groups of order 2.
Find the multiplication tables of all groups of order 3.
Find the multiplication tables of all groups of order 4.
Find the multiplication tables of all groups of order 5.
Write the permutation 12, 23, 31, in diagram notation, in two line notation, in cycle notation, and in matrix notation.
Write the permutation 13, 22, 31, in diagram notation, in two line notation, in cycle notation, and in matrix notation.
Write the permutation 13, 24, 35, 42, 51, in diagram notation, in two line notation, in cycle notation, and in matrix notation.
Calculate the products ( 1 2 3 3 2 1 ) ( 1 2 3 2 3 1 ) and ( 1 2 3 4 3 1 2 4 ) ( 1 2 3 4 2 3 4 1 ).
Write all elements of the symmetric group S3 in diagram notation, in two line notation, in cycle notation, and in matrix notation.
Determine whether the set of positive real numbers with the operation of addition is a group.
Determine whether the set of n×n matrices over the real numbers with the operation of multiplication is a group.
Let G be a group and let x,y G. Show that there exist unique w and z in G such that wx=y and xz= y. Is w=z?
Show that the set {z | n>0 ,zn=1} forms a group under multiplication.
Compute the following products of permutations: (123) (456) * (134) (25) (6), (12345) * (1234567) and (123456)* (123)* (123)* (1) .
Let X=-{0,1}. Show that the following functions from X to X with the operation of composition of functions form a group: f= 11-x, g= x-1x, h= 1x, i=x, j=1-x, k= xx-1.
Show that 2 is a subgroup of the group .
Show that the set of negative integers is not a subgroup of the group .
Let G be a group and let 1G. Show that {1} is a subgroup of G.
Show that { (1), (123), (132) } is a subgroup of S3.
Show that { (1), (12), (23), (13) } is not a subgroup of S3.
Let G be a group and let gG. Show that {gn | n} is a subgroup of G.
Let n>0. Calculate the order of /n, Always justify your answers.
Let n>0. Calculate the order of Sn. Always justify your answers.
Calculate the orders of the elements of /12, Always justify your answers.
Calculate the orders of the elements of S4, Always justify your answers.
Show that S3 is nonabelian and noncyclic.
Define the function exp: >0 and prove that it is a group isomorphism.
Prove that /4 and /2× /2 are nonisomorphic groups of order 4.
Prove that /6 and /2× /3 are isomorphic groups of order 6.
Prove that /6 and S3 are nonisomorphic groups of order 6.
Prove that the groups /8 and /4× /2 and /2× /2× /2 are all nonisomorphic.
Calculate the order of On().
Calculate the order of SOn().
Find the order of the element (123) (4567) (89) in S10.
Find the order of the element (14) (23567) in S7.
Find the orders of the elements 6, 12, 11, and 14 in /20.
Find the orders of the elements 2, 12 and 8 in /13.
Let G be a group and let gG. Show that the order of g is equal to the order of g-1.
Let G be a commutative group and let g ,hG. Show that if g and h have finite order then gh has finite order.

References

[GH] J.R.J. Groves and C.D. Hodgson, Notes for 620-297: Group Theory and Linear Algebra, 2009.

[Ra] A. Ram, Notes in abstract algebra, University of Wisconsin, Madison 1994.