Abstract Algebra

Questions and Solutions on Permutation

Group
P. Kalika∗& K. Munesh†
August 13, 2015

Questions related to Permutation Group

1. Illustrate Cayley’s Theorem by calculating the left regular representation for the
group V4 = {e, a, b, c} where a2 = b2 = c2 = e, ab = ba = c, ac = ca = b, bc =
cb = a.

2. Show that A5 has 24 elements of order 5, 20 elements of order 3, and 15 elements

of order 2.

3. Show that if n ≥ m then the number of m-cycles in Sn is given by n(n − 1)(n −

2)...(n − m + 1)/m.

4. Let σ be the m-cycle (12 . . . m). Show that σ i is also an m-cycle if and only if i
is relatively prime to m.

5. Let n ≥ 3. Prove the following in Sn .

(a) Every permutation of Sn can be written as a product of at most n − 1

transpositions.
(b) Every permutation of Sn that is not a cycle can be written as a product of
at most n − 2 transpositions.

6. Let σ be a permutation of a set A. We say that σ moves a ∈ A if σ(a) 6= a. Let

SA denote the permutations on A.

(a) If A is a finite set then how many elements are moved by a n-cycle σ ∈ SA ?
∗
E-mail: klkaprsd@gmail.com, M.Sc Tech Mathematics, Central University of Rajasthan
†
M.Sc Mathematics, Central University of Rajasthan

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

(b) Let A be an infinite set and let H be the subset of SA consisting of all σ ∈ SA
such that σ only moves finitely many elements of A. Show that H ≤ SA .
(c) Let A be an infinite set and let K be the subset of SA consisting of all
σ ∈ SA such that σ moves at most 50 elements of A. Is K ≤ SA ? Why?

7. Show that if σ is a cycle of odd length then σ 2 is a cycle.

8. Let p be a prime. Show that an element has order p in Sn if and only if its cycle
decomposition is a product of commuting p-cycles. Show by an explicit example
that this need not be the case if p is not prime.

9. Show that if n ≥ 4 then the number of permutations in Sn which are the product
of two disjoint 2-cycles is n(n − 1)(n − 2)(n − 3)/8.

10. Let b ∈ S7 and suppose b4 = (2143567). Find b.

11. Let b = (123)(145). Write b99 in disjoint cycle form.

12. Find three elements σ in S9 with the property that σ 3 = (157)(283)(469).

13. Show that if H is a subgroup of Sn , then either every member of H is an even

permutation or exactly half of the members are even.

14. Suppose that H is a subgroup of Sn of odd order. Prove that H is a subgroup of

An .

15. Prove that the smallest subgroup of Sn containing (12) and (12 . . . n) is Sn . In
other words, these generate Sn .

16. Prove that for n ≥ 3 the subgroup generated by the 3-cycles is An .

17. Prove that if a normal subgroup of An contains even a single 3-cycle it must be
all of An .

18. Prove that A5 has no non-trivial proper normal subgroups. In other words show
that A5 is a simple group.

19. Show that Z(Sn ) is trivial for n ≥ 3.

20. Show that two permutations in Sn are conjugate if and only if they have the
same cycle structure or decomposition. Given the permutation x = (12)(34),
y = (56)(13), find a permutation a such that a−1 xa = y.

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

Solution of above problems

1. Illustrate Cayley’s Theorem by calculating the left regular representation for the
group V4 = {e, a, b, c} where a2 = b2 = c2 = e, ab = ba = c, ac = ca = b, bc =
cb = a.
Solution :
Let V4 = {e, a, b, c}. Now computing the permutation σg induced by the action
of left-multiplication by the group element a.
a.e = ae = a and so σg (e) = a
2
a.a = aa = a = e and so σg (a) = e
a.b = ab = c and so σg (b) = c
a.c = ac = b and so σg (c) = b
Hence σa = (ea)(bc).

Now computing σg induced by the action of left-multiplication by the group

element b.
b.e = be = b and so σg (e) = b
b.a = ba = c and so σg (a) = c
b.b = bb = b2 = e and so σg (b) = e
b.c = bc = a and so σg (c) = a
Hence σb = (eb)(ac).

Similarlly Computing σg induced by the action of left-multiplication by the group

element c.
c.e = ce = c and so σg (e) = c
c.a = ca = b and so σg (a) = b
c.b = cb = a and so σg (b) = a
c.c = cc = c2 = e and so σg (c) = e
Hence σc = (ec)(ab).

Which explicitly gives the permutation representation V4 → V4 associated to

this action.
2. Show that A5 has 24 elements of order 5, 20 elements of order 3, and 15 elements
of order 2.
Solution :
Since we can decompose any permutation into a product of disjoint cycle. In S5 ,
since disjoint cycle commutes. Let V5 = {e, a, b, c, d} Here an element of S5 must
have one the following forms:
(i) (abcde) - even
(ii) (abc)(de) - odd (even P * odd P)
(iii) (abc) - even

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

(iv) (ab)(cd) - even (odd P * odd P)

(v) (ab) - odd
(vi) (e) -even
So element of A5 is of the form (i), (iii), (iv) and (vi). As we know that, when
a permutation is written as disjoint cycles, it’s order is the lcm (least common
multiple) of the lengths of the cycles.
(i) (abcde) has order 5
(iii) (abc) has order 3
(iv) (ab)(cd) has order 2
(vi) (e) has order 1
Now since elements of order 5 in A5 are of the form (i). There are 5! distinct
expression for cycle of the form (abcde) where all a, b, c, d, e are distinct. since
expression representation of the element of type
(abcde) = (bcdea) = (cdeab) = (deabc) = (eabcd) are equivalent. So total ele-
5×4×3×2×1
ments of order 5 are = 24.
5
Now for elements of order 3. Since elements of order 3 in A5 is of the form (abc).
Here there are 5 choices for a, 4 choices for b and 3 choices for c. so there are
5 × 4 × 3 = 60 possible ways to write such a cycle. Since expression representa-
tion of the element of type (abc) = (bca) = (cab) are equivalent.So total no. of
60
elements of order 3 in A5 are = 20.
3
Here since even permutation of order 2 are of the form (ab)(cd). so there are
5 × 4 × 3 × 2 ways to write such permutation. Since disjoint cycles commute
there, so there are 8 different ways that differently represent the same permuta-
tions :-
(ab)(cd) = (ab)(dc) = (ba)(dc) = (ba)(cd) = (cd)(ab) = (dc)(ab) = (dc)(ba) =
(cd)(ba).
5×4×3×2
So there are = 15 elements of order 2.
8
{No. of ways of selecting r different things out of n is nP r }

3. Show that if n ≥ m then the number of m-cycles in Sn is given by

n(n − 1)(n − 2)...(n − m + 1)
.
m
Proof :
For any given Sn , there are n elements in Sn = {1, 2, 3, ...m...n}. so we must have
n-choices for 1st element, then n-1 choices for 2nd element, n-2 choices for 3rd
element and so on... and we have n-m+1 choices for mth element etc. So there
are total no. of n(n-1)(n-2)...(n-m+1) for a m-cycles.
Now we want to count m-cycles in Sn , since for 2-cycles (ab) = (ba)
{two equivalent notation , i.e same permutation}

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

For 3-cycles (a, b, c) = (b, c, a) = (c, a, b) {i.e 3-equivalent notation}

For 4-cycles (a, b, c, d) = (b, c, d, a) = (c, d, a, b) = (d, a, b, c) {four equivalent
notation}
-----------
Similarly for m-cycles there are m-equivalent notation for any permutations.
Now, Since we have, n(n − 1)(n − 2)...(n − m + 1) choices to form a m-cycle in
which there are m-equivalent notations for any permutation of length m.
So the no. of m-cycles in Sn is

n(n − 1)(n − 2)...(n − m + 1)

m

4. Let σ be the m-cycle (12 . . . m). Show that σ i is also an m-cycle if and only if i
is relatively prime to m.
Proof :
First we note that if τ is k cycle then |τ | = k
since σ i (x) ≡ x+i mod m for any x, 1 ≤ x ≤ m
Claim : σ i = (σ i (1)σ i (2)...σ i (m))
we prove it by contradiction
Let i=1. Then the statement is obviously true.
Suppose that

σ i−1 = (σ i−1 (1)σ i−1 (2)...σ i−1 (m))

then σ i = σ(σ i−1 ) = σ{σ i−1 (1)...σ i−1 (m)}

Since, here σ sends σ i−1 (i) to σ i (1),
thus σ i = (σ i−1 (1)...σ i (m))
=⇒ σ i = (σ i−1 (1)...σ i (m))
Since σ i (m) ≡ m+i mod m ≡ i mod m and σ i−1 (1) ≡ 1+i-1 mod m ≡ i mod m
i.e σ i (m) = σ i−1 (1)
=⇒ σ i is an m-cycle.
Converse part
Suppose σ i is an m-cycle and suppose that (i, m) = d > 1. (we prove it by
contradiction)
then there exists k,n ∈ N such that i=kd and m=nd,
since, (σ i )n = (σ kd )n = σ kdn = σ mk = (σ m )k = I
where I is the identity permutation.
Hence |σ i | ≤ n < m.
which is contradiction, since σ i is an m-cycle and thus |σ i | = m. Thus i is
relatively prime to m.

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

5. Que. No.05 Let n ≥ 3. Prove the following in Sn .

(a) Every permutation of Sn can be written as a product of at most n − 1

transpositions.
(b) Every permutation of Sn that is not a cycle can be written as a product of
at most n − 2 transpositions.

Proof (a) :
We know that if k ≥ 2, the cycle (a1 , a2 , ...ak ) can be written as (a1 , ak )(a1 , ak−1 )...(a1 , a2 )
which is k-1 transpositions.
Case-I, If k=1, then this cycle is the trivial cycle or the identity, which can be
written as 1-1=0 transpositions
Case-II, if k > 1,
we know that every permutation σ ∈ Sn can be written as a product of disjoint
cycles, thus we can write
σ = (a11 , a12 , ..., a1k1 )(a21 , a22 , ..., a2k2 )...(am1 , am2 , ..., amkm )
where k1 + k2 + ... + km = n and each of these cycle is disjoint.
we know that cycle Pm i canPbe written as a product of ki − 1 transpositions, and
P m m
i=1 (ki −1) = i=1 ki − i=1 1 = n−m, this is maximized when m is minimized
and the least value of m is 1.
Thus, the largest value of n-m can be n-1.
Proof (b) :
From part (a), σ = (a11 , a12 , ..., a1k1 )(a21 , a22 , ..., a2k2 )...(am1 , am2 , ..., amkm ) where
P m
i=1 ki = n and each of cycles is disjoint and also from (a), we still know that
cycles
Pm i can be written
Pm as a P product of ki − 1 transpositions and
m
(k
i=1 i − 1) = k
i=1 i − i=1 1 = n − m, However, since σ is not a cycle.
m ≥ 2, thus n-m is maximized when m is minimized i.e m=2 i.e n-2 is the maxi-
mum value of n-m.
Hence every permutation of Sn that is not a cycle can be written as a product of
at most n-2 transpositions.

6. Que. No.06 Let σ be a permutation of a set A. We say that σ moves a ∈ A if

σ(a) 6= a. Let SA denote the permutations on A.

(a) If A is a finite set then how many elements are moved by a n-cycle σ ∈ SA ?
(b) Let A be an infinite set and let H be the subset of SA consisting of all σ ∈ SA
such that σ only moves finitely many elements of A. Show that H ≤ SA .
(c) Let A be an infinite set and let K be the subset of SA consisting of all
σ ∈ SA such that σ moves at most 50 elements of A. Is K ≤ SA ? Why?

Proof (a):
If A is finite, then σ moves only n elements because σ is n-cycle and the elements

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

which is not in cycle are fixed.

Proof (b):
We may prove it by One-Step Subgroup Test.
As A is infinite set and σ ∈ SA moves only finitely many elements of A. Since H
consists all σ ∈ SA
⇒ H is non-empty.
Now let, σ ∈ H =⇒ σ −1 ∈ H.
So, σoσ −1 = I =∈ H
Now checking for closure property,
Let σ1 and σ2 ∈ H be any two permutations such that σ1 and σ2 both moves
only finitely many elements of A.
Then σ1 oσ2 also moves only finitely many elements of A.
⇒ Closure property holds.
⇒ H is subgroup of A5 .
Proof (c):
No, K will not be subgroup of SA
Because, suppose that σ1 moves at most 50 elements and σ2 moves at most 50
elements, then σ1 oσ2 (Product of two permutations) might moves more than 50
elements.
⇒ Closure property with respect to function composition is not satisfied in K.
⇒ K is not a subgroup of SA .
7. Que. No.07 Show that if σ is a cycle of odd length then σ 2 is a cycle.
Proof : Suppose σ : A → A is a cycle with odd length. Then we can write σ
in a cycle notation as σ
σ = (a1 , a2 , ..., aak+1 ) where a1 , a2 , ..., a2k+1 ∈ A
On simple calculation, we may show that
σ 2 = (a1 , a2 , ...a2k+1 )(a1 , a2 , ...a2k+1 )
σ 2 = (a1 , a3 , a5 , ...a2k+1 , a2 , a4 ...a2k )
=⇒ σ 2 is cycle whenever σ is cycle.
8. Que. No.08 Let p be a prime. Show that an element has order p in Sn if and
only if its cycle decomposition is a product of commuting p-cycles. Show by an
explicit example that this need not be the case if p is not prime.
Proof :
⇒ Suppose the order of σ is p(p is prime).
Since order of σ is the lcm of the sizes of the disjoint cycles in the cycle decom-
position of σ, So all of these cycle must have sizes that divides p is either 1 or
p.
Since 1-cycles are omitted from the notation for the cycle decomposition of σ.
Thus the cycle decomposition consists entirely of p-cycles. Thus σ is the product
of disjoint commuting p-cycles.

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

⇐ Suppose σ is the product of disjoint p-cycles. i.e σ = c1 c2 c3 ...cr

then σ p = (c1 c2 c3 ...cr )2 = cp1 cp2 cp3 ...cpr = 1
(since the pth power of p-cycles in σ are all 1, so their product is 1)
σp = 1
A p-cycle has order p, so no smaller power of σ can be 1. Hence |σ| = p.
For an example :
Showing these conclusions may fail when p is not a prime.
Let p=6, σ = (12)(345)
|σ| = lcm(2, 3) = 6
but σ is not the product of commuting 6-cycles.

9. Que. No.09 Show that if n ≥ 4 then the number of permutations in Sn which

are the product of two disjoint 2-cycles is n(n − 1)(n − 2)(n − 3)/8.
Solution :
Given n≥ 4.
Since, Permutations which are the product of two disjoint 2-cycles is of the form
(ab)(cd), i.e of length 4.
Hence, there are n choices for a, (n-1) choices for b, (n-2) choices for c and (n-3)
choices for d.
So there are n(n − 1)(n − 2)(n − 3) possible ways to write to write such a cycle.
Since disjoint cycles commutes there, so there are 8 different ways that differently
represent the same cycle(As i mentioned it in sol. of Que.2)
Hence total number of Permutation in Sn which are the product of two disjoint
(n)(n − 1)(n − 2)(n − 3)
2-cyles is .
8

10. Que. No.10 Let b ∈ S7 and suppose b4 = (2143567). Find b.

Solution :

∵ b ∈ S7
|b| = 7
⇒ b7 = I
So b = Ib = (b7 ).b = b8 = (b4 )2
⇒ b = b4 .b4
⇒ b = (2143567)(2143567)
= (2457136).

As given that b4 = (2143567).

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

11. Que. No.11 Let b = (123)(145). Write b99 in disjoint cycle form.
Solution :
Since b = (123)(145) = (14523). So order of b is 5.
(In case of single cycle. The order of permutation is the degree of permutation
is the lengths of the set.)
Now since |b| = 5, then b5 = I.
So we can write b99 = (b5 )19 .b4 = Ib4 = b4 = b−1 .
Since b = (14523) ⇒ b4 = b−1 = (32541) = (132541)
so b99 = (13254) or (154)(132).

12. Que. No.12 Find three elements σ in S9 with the property that σ 3 = (157)(283)(469).
Solution :
Let 1 = a1 , 2 = a2 , 3 = a3 , 4 = a4 , 5 = a5 , 6 = a6 , 7 = a7 and 8 = a8 .
Now we have to find σ such that σ 3 = (a1 a5 a7 )(a2 a8 a3 )(a4 a6 a9 )
then σ1 = (a1 .... a5 .... a7 .... )
σ1 = (a1 a2 .. a5 a8 .. a7 a3 .. )
σ1 = (a1 a2 a4 a5 a8 a6 a7 a3 a9 )
σ1 = (1 2 4 5 8 6 7 3 9).
Similarly we can find other two elements
σ2 = (a1 .... a5 .... a7 .... )
σ2 = (a1 a3 .. a5 a2 .. a7 a8 .. )
σ2 = (a1 a3 a9 a5 a2 a4 a7 a8 a6 )
σ2 = (1 3 9 5 2 4 7 8 6).
and
σ3 = (a2 .... a8 .... a3 .... )
σ3 = (a2 a1 a4 a8 a5 a6 a3 a7 a9 )
σ3 = (2 1 4 8 5 6 3 7 9).

13. Que. No.13 Show that if H is a subgroup of Sn , then either every member of
H is an even permutation or exactly half of the members are even.
Solution :
Let H ⊂ Sn be any subgroup.
Now, we define H = {σ ∈ H — σ is even }
Claim: H is subgroup of H.
Let f,g ∈ H, Since g are even, so g −1 is also even.
since the product of even permutations are still even, so we have f og −1 is even.
So, here there are only two possibilities either H = H or H $ H
Case-I, if H = H, then we are done.

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

|H|
Case-II, if H 6= H, then we need to show that |H| =
2
Since H 6= H ,it implies that there exists at least one odd permutations σ ∈ H
H
Now consider f: H → defined by f(h) = σ.h for any h ∈ H.
H
since σ is odd and h is even
⇒ σ.h is odd.
H
⇒ σ.h ∈
H
To prove that H = |H| 2
, We need to prove f is 1-1 and onto.
for 1-1
let h1 , h2 ∈ H such that h1 = h2.
since h1 = h2
⇒ σh1 = σh2 ⇒ f (h1 ) = f (h2 ) ⇒ f is 1-1.
and for onto
H
since f −1 : HH
→ H is given by f −1 (h) = σ −1 h0 for every h’ ∈ .
H
So f is both 1-1 and onto
H |H|
⇒ |H| = | |, hence |H| =
H 2

14. Que. No.14 Suppose that H is a subgroup of Sn of odd order. Prove that H is
a subgroup of An . rate Sn .
Proof :
Let H be a subgroup of Sn of odd order.
i.e |H| = odd order
We may prove it by contradiction.
To the contrary, suppose H * An , then
suppose ∃ σ ∈ H such that σ is an odd permutation.
Let H = {α1 , α2 , α3 , ...., αp } ∪ {β1 , β2 , β3 , ..., βq }
| {z } | {z }
Odd Even

∴ σH = {σα1 , σα2 , σα3 , ..., σαp } ∪ {σβ1 , σβ2 , σβ3 , ..., σβq }
| {z } | {z }
Even Odd

=H
=⇒ p = q
=⇒ |H| = 2p = 2q = even
Which is a contradiction.
=⇒ H ⊂ An
15. Que. No.15 Prove that the smallest subgroup of Sn containing (12) and (12 . . . n)
is Sn . In other words, these generate Sn .
Proof :
Let σ = (12) and τ = (123...n)

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

Suppose H is subgroup of Sn which contains both σ = (12) and τ = (123...n).

Now, we need to show that H = Sn .
Clearly, we have H ⊆ Sn . Since subgroups in particular are subsets.
Since we know that Sn is generated by (n-1) transpositions (12)(23)(34)(45)...(n-
1 n).
Now, I want to show that (12) and (123...n) generates these (n-1) transposition.
Consider, τ στ −1
(12...n)(12)(12...n)−1 = (23)
(12...n)(23)(12...n)−1 = (34)
(12...n)(34)(12...n)−1 = (45)
. . . . . . . . . . .
(12...n)(n − 2 n − 1)(12...n)−1 = (n − 1 n)
(12...n)(n − 1 n)(12...n)−1 = (n 1)

Now i prove it by induction...

for n = 1, it is obviously true.
We assume that it is true for n = k, then
(12...k)(k − 1 k)(12...k)−1 = (k 1)
Now, we wish to show that it is true for n = k+1
(1, 2, ..., k, k + 1)(k, k + 1)(1, 2, ..., k, k + 1)−1
= (1, 2, ..., k, k + 1)(k + 1, k)(k + 1, k, ..., 3, 2, 1)
= 6(1, 2, ..., k, k + 1)(k + 1)(k, ..., 3, 2, 1)
= (1, 2, ..., k, k + 1)(k, ..., 3, 2, 1)
= (k)(k − 1)...(3)(2)(1)(1, k + 1)
= (k+1, 1)
So, it is true for n=k+1
⇒ (12) and (123...n) generates Sn
Which shows that Sn ⊆ H.
Thus h = Sn

16. Que. No.16 Prove that for n ≥ 3 the subgroup generated by the 3-cycles is An .
Proof :
Since every 3-cycle is an even permutation, then every 3-cycle of Sn is in An .
Now, Let τ ∈ An ⇒ τ is an even permutation.
⇒ τ is a product of an even no. of transposition.
However, (a1 a2 )(a3 a4 ) = (a1 a2 a3 )(a2 a3 a4 )
And (a1 a2 )(a1 a3 ) = (a1 a3 a2 )
Consequently, every product of two transposition(whether they share an element
or not) can be written as a product of 3-cycles.
Hence, τ can be written as a product of 3-cycles.
⇒ For any n≥ 3, the subgroup generated by 3-cycle is An .

17. Que. No.17 Prove that if a normal subgroup of An contains even a single 3-cycle

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

it must be all of An .
Proof :
Let N ⊂ An be Normal subgroup and suppose that (abc) ∈ N. Let σ 0 ∈ An be
an arbitrary 3-cycles.
Then σ 0 = τ (abc)τ −1 for some τ ∈ Sn .
Now here, there are two possibility either τ ∈ An or τ ∈ / An .
0
Case -I, If τ ∈ An then σ ∈ N and we are done.
Case -II, If τ ∈ / An then τ 0 = τ (ab) is in An and τ 0 = τ (acb)τ 0−1 is once again in
N.
⇒ If N E An and contains a 3-cycle. Then N=An .

18. Que. No.18 Prove that A5 has no non-trivial proper normal subgroups. In
other words show that A5 is a simple group.
Solution :
5!
Order of A5 = |A5 | = = 60 = 22 .3.5.
2
Let N be proper normal subgroup of A5 , then
|N | = 2 , 3 , 4 , 5 , 6 , 10 , 12 , 15 , 20 , 30.
5P 5
Total no. of 5 order elements in A5 = = 24,
5
5P 3
Total no. of elements of 3 order in A5 = = 20,
5
And total no. of 15-order elements in A5 = 0.
Let us assume that |H| = 3 , 6 , 12 , 15
A5
then | | = 20 , 10 , 5 , 4
H  
A5
so gcd 3 , | | = 1
H
=⇒ H would contain all 20 elements of order 3.
Which is a contradiction.
{ As, Theorem  says that If H be Normal subgroup of a finite group G. And if
G
gcd |x|, | | =1, then x ∈ G}.
H
Similarly, suppose that |H| = 5 ,10 , 20
A5
then | | = 12 , 6 , 3
H
=⇒ H would contain all 24 elements of order 5.
which is a contradiction.
A5
Let |H| = 30, then | | = 2.
 H  
A5 A5
So again gcd 3 , | | = 1 and gcd 5 , | | = 1.
H H
=⇒ H would contain all 20+24 = 44 elements.
we get again a contradiction.

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

And finally, let us assume that, |H|= 2 or 4.

A5
=⇒ | | = 30, 15
H
Since, we know that any group of order 30 or 15 has an element of order 15.
A5
or As, if | | = 15 = 3 ×5 = p × q where p=3 and q=5.
H
( Theorem : If G is a group of order pq, where p and q are primes, p < q and
p - q, then G is cyclic.)
⇒ G has at least one element of order 15.
Which is again contradiction,
A5
because A5 contains no such element, neither does .
H
This proves that A5 is simple.

19. Que. No.19 Show that Z(Sn ) is trivial for n ≥ 3.

Solution :
Let σ ∈ Sn be a non-identity element then there exists two distinct a,b ∈
{1, 2, 3, ..., n} with σ(a) = b.
Since n ≥ 3, Now choosing k ∈ {1, 2, 3, ..., n} such that k 6= a and k 6= b.
Let τ = (ak). Then
τ (σ(a)) = τ (b) = k and σ(τ (a)) = σ(a) = b
since k6=b ⇒ τ (σ(a)) 6= σ(τ (a)).
Hence for every non-identity permutation in Sn , there exists some element not
commuting with it.
Therefore Z(Sn ) must be trivial.

20. Que. No. 20 Show that two permutations in Sn are conjugate if and only if
they have the same cycle structure or decomposition. Given the permutation
x = (12)(34), y = (56)(13), find a permutation a such that a−1 xa = y.
Proof :
For any σ and any d ≤ n, we have
σ(12...d)σ −1 = (σ(1)σ(2)....σ(d))
This shows that any conjugate of d-cycle is again d-cycle.
Since every permutation is a product of disjoint cycles, it follows that the cycle
structure of conjugate permutations are the same.
In other direction,
Let τ = (a1 a2 .....ar )(ar+1 ar+2 .....as )....(al .....am ) and
τ 0 = (a01 a02 .....ar )(ar+1 ar+2 .....as )....(al .....am )
be two permutations having the same cycle structure.
Define σ ∈ Sn by σ(a0i ) = a0 for i = 1,2,...,m then
στ σ −1 = σ(a1 a2 ....ar )σ −1 σ(ar+1 ar+2 ....as )σ −1 ....σ(al ....am )σ −1
= (a01 a02 .....ar )(ar+1 ar+2 .....as )....(al .....am )

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P. Kalika & K. Munesh Permutation Group Que. & Sol.

= τ0
This shows that τ and τ 0 are conjugate.
Now, Given the permutation x = (12)(34), y = (56)(13)
Since that a−1 xa = y.
∴ xa = ay ⇒ x = aya−1 .
⇒ ((12)(34)) = a((56)(13))a−1
⇒ ((12)(34))(5)(6) = a((56)(13)(2)(4))a−1
. = (a(5)a(6))(a(1)a(3))a(2)a(4)
⇒ 1 = a(5), 2 = a(6), 3 = a(1), 4 = a(3) and 5 = a(2), 6 = a(4)

1 2 3 4 5 6
⇒a=
3 5 4 6 1 2
⇒ a = (134625)
Checking for a, a = (134625) and a−1 = (526431) = (152643)
∴ a−1 xa = (134625)((12)(34))(152643)
= (13)(2)(4)(56) = (13)(56) = RHS, Hence done.

Acknowledgement
This problem & solution is part of my INSPIRE(DST) summer work, which has been
done at Ambekar University, Delhi. I thank my supervisor Prof. Geetha Venkataraman
who gave me this opportunity for this summer internship and also for her fruitful
guidelines and discussion. I also thank DST(Department of Science and Technology)
Govt. of India, India for the financial Grant.

References
[1] Joseph A. Gallian : Contemporary Abstract Algebra, Ch-5, Brooks/Cole, Cengage
Learning, ISBN: 978-0-547-16509-7, 7th Ed. (2010)

[2] VK Khanna & SK Bhamri : A course in abstract algebra, 1998

[3] David S. Dummit & Richard M. Foote : Abstract Algebra, Ch-1, John Wiley &
Sons, Inc, ISBN: 0-471-43334-9, 3rd Ed. (2004).

[4] I. N. Herstein : Topics in Algebra, John Wiley & Sons, Ch-2, 2nd Ed (1975).

[5] John B. Fraleigh : A First Course in Abstract Algebra

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