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Symmetric groups and Permutation groups

Modern Algebra
Group theory
Symmetric Groups
Permutation groups

1. This question is concerned with subgroups of the group S5 of symmetries (or permutations)
on the set {1,2,3,4,5}, a group with 120 elements.

(a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6 and give examples of
each of these.
Explain why this group does not have cyclic subgroups of any other order.

(b) By considering the symmetry groups of appropriate geometric figures, give examples of :
(i) a subgroup of order 4 that is not cyclic;
(ii) a subgroup of order 6 that is not cyclic;
(iii) a subgroup of order 8.

(c) By considering those permutations that fix one element, or, otherwise, give an example of
a subgroup of order 24 and another of order 12. [You need not list all the elements of these
groups, but you should explain clearly which elements constitute each subgroup.]

(d) List the potential orders of subgroups of S5 (other than S5 itself), according to Lagrange's
theorem, in addition to those already considered in this question.
Give an example of a subgroup of one of these orders.

2. (a) Which of the following sets are groups under the specified binary operation? In each case,
justify your answer.
(i) Z, the set of integers, under operations * defined by a*b = a + 2b
(ii) R*, the set of non-zero real numbers, under the operation × defined by x×y = 5xy
(iii) The set {3,6,9,12} under multiplication modulo 15.
(iv) The set of matrices {(1, p;0,1)/pЄZ} under matrix multiplication.

(b) G is a group of real functions with domain and co-domain the non-negative real numbers,
i.e. functions [0,∞) → [0,∞). The group operation in G is function composition.
If one of the elements of G is the squaring function, f, defined by f(x) = x2 , explain why G
must be an infinite group.

4. (a) Define the motion of conjugacy as it applies in a general group.
Prove that the inverses of a pair of conjugate elements are also conjugate.
Prove that conjugate elements have the same order.
The remainder of this question concerns the group G, whose Cayley table is as follows:

e a b c d f g h i j k l
----|--------------------------------------------------------------------------------------------------
e | e a b c d f g h i j k l
a | a b e d f c h i g k l j
b | b e a f c d i g h l j k
c | c f d e b a j l k g i h
d | d c f a e b k j l h g i
f | f d c b a e l k j i h g
g | g h i j k l e a b c d f
h | h i g k l j a b e d f c
i | i g h l j k b e a f c d
j | j l k g i h c f d e b a
k | k j l h g i d c f a e b
l | l k j i h g f d c b a e

(b) Determine the inverse and the order of each of the elements of G.

(c) Simplify each of the following:

(i) acb
(ii) bca
(iii) ajb
(iv) bja
(v) gcg

(d) Given that the only element conjugate to g is g itself ( you need not prove this),
determine the conjugacy classes of G.

(e) Find H, a normal subgroup of G having three elements. Identify the elements of the
quotient group G/H and determine its isomorphism type.

See the attached files.

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Solution Preview

Modern Algebra
Group theory
Symmetric Groups
Permutation groups

1. This question is concerned with subgroups of the group S5 of symmetries (or permutations)
on the set {1,2,3,4,5}, a group with 120 elements.

(a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6 and give examples of
each of these.
Explain why this group does not have cyclic subgroups of any other order.

(b) By considering the symmetry groups of appropriate geometric figures, give examples of :
(i) a subgroup of order 4 that is not cyclic;
(ii) a subgroup of order 6 that is not cyclic;
(iii) a subgroup of order 8.

(c) By considering those permutations that fix one element, or, otherwise, give an example of
a subgroup of ...

Solution Summary

This solution is comprised of a detailed explanation of the problems of Symmetric groups and Permutation groups.
It contains step-by-step explanation for the following problems:

1. This question is concerned with subgroups of the group S5 of symmetries (or permutations)
on the set {1,2,3,4,5}, a group with 120 elements.

(a) Explain why this group has cyclic subgroups of order 1,2,3,4,5 and 6 and give examples of
each of these.
Explain why this group does not have cyclic subgroups of any order.

(b) By considering the symmetry groups of appropriate geometric figures, give examples of :
(i) a subgroup of order 4 that is not cyclic;
(ii) a subgroup of order 6 that is not cyclic;
(iii) a subgroup of order 8.

(c) By considering those permutations that fix one element, or, otherwise, give an example of
a subgroup of order 24 and another of order 12. [You need not list all the elements of these
groups, but you should explain clearly which elements constitute each subgroup.]

(d) List the potential orders of subgroups of S5 (other than S5 itself), according to Lagrange's
theorem, in addition to those already considered in this question.
Give an example of a subgroup of one of these orders.

2. (a) Which of the following sets are groups under the specified binary operation? In each case,
justify your answer.
(v) Z, the set of integers, under operations * defined by a*b = a + 2b
(vi) R*, the set of non-zero real numbers, under the operation × defined by x×y = 5xy
(vii) The set {3,6,9,12} under multiplication modulo 15.
(viii) The set of matrices {(1, p;0,1)/pЄZ} under matrix multiplication.

(b) G is a group of real functions with domain and co-domain the non-negative real numbers,
i.e. functions [0,∞) → [0,∞). The group operation in G is function composition.
If one of the elements of G is the squaring function, f, defined by f(x) = x2 , explain why G
must be an infinite group.

4. (a) Define the motion of conjugacy as it applies in a general group.
Prove that the inverses of a pair of conjugate elements are also conjugate.
Prove that conjugate elements have the same order.
The remainder of this question concerns the group G, whose Cayley table is as follows:

e a b c d f g h i j k l
----|--------------------------------------------------------------------------------------------------
e | e a b c d f g h i j k l
a | a b e d f c h i g k l j
b | b e a f c d i g h l j k
c | c f d e b a j l k g i h
d | d c f a e b k j l h g i
f | f d c b a e l k j i h g
g | g h i j k l e a b c d f
h | h i g k l j a b e d f c
i | i g h l j k b e a f c d
j | j l k g i h c f d e b a
k | k j l h g i d c f a e b
l | l k j i h g f d c b a e

(b) Determine the inverse and the order of each of the elements of G.

(c) Simplify each of the following:

a. acb
b. bca
c. ajb
d. bja
e. gcg

(d) Given that the only element conjugate to g is g itself ( you need not prove this),
determine the conjugacy classes of G.

(e) Find H, a normal subgroup of G having three elements. Identify the elements of the
quotient group G/H and determine its isomorphism type.

Solution contains detailed step-by-step explanation.

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