1. ## Group theory problems

Group Theory(Nov.2006)

1.a)If phi:G ->G' is a homomorphism of the group G into the group G' then with usual notations prove that

i)phi(1)=1

ii)phi(xn)=[phi(x)]n, x belongs to G, n belongs to Z

iii)phi preserves all group relations.

b)If G is of finite order n then prove that the group of automorphisms of G (i.e.AutG) is isomorphic to the group of those positive integers <n, which are co-prime to n, with binary operation as multiplication modulo n and order of Aut(G)=phi(n)(phi is Euler's phi function).

Hence or otherwise show that

Aut(Zn,+n)is isomorphic to (Zncross,cross).

c)Construct an abelian group G of order 27 in which x3 = 1 for all x in G.

2.a)i)Define a cycle of length l(i.e.l-cycle).

Prove that number of l-cycles in Sn is {(n)(n-1)...(n-l+1)}/l

ii) If sigma is a cycle of length l, then prove that tsigmat-1 is also a cycle of length l for any t belongs to Sn.

b)i) Define symmetric polynomial in n variables x1,x2,...xn.

Determine which of the following polynomial is symmetric polynomial

I)f(x, y, z)= x2y+y2z+z2x

II)g(x, y, z)= x2y + xy2 + y2z + yz2 + z2x + zx2.

ii)Find the group of symmetries of the polynomial f(x1,x2,x3)=product of(xi -xj), i<j.

c)Determine the elements of order n in ccross, the group of non-zero complex numbers under multiplication.

3.a)Define the cyclic structure of element sigma in Sn.

Prove that two elements of Sn are conjugate to each other if and only if they have the same cyclic structure.

b)Determine the conjugacy classes in S3 and A3 and hence write down their class equation.

c)i)Show that in any group G, ab and ba are conjugate to each other. What can you say about the orders of ab and ba ?

ii)Rule out as many of the following as possible as class equations for a group of order 10.

1+1+1+2+5, 1+2+2+5, 1+2+3+4, 1+1+2+2+2+2.

4.a)i)Show that a simple abelian group is cyclic of prime order.

ii)Show that for K=Z,Q,R or C, SL(n,K) is a normal subgroup of GL(n,K).

b)State Cauchy's theorem for finite abelian group. Use this to prove that the converse of Lagrange's theorem for finite abelian group holds.

c)Show that a group of order 4 is either cyclic of isomorphic to V4.

5.a)If N,H are subgroups of a group G such that N is normal in G then prove that HN/N is isomorphic to H/(H intersection with N)

b)If N and C are respectively the normalizer and centralizer of a subgroup H of a group G, then show that C is normal in N and that N/C is isomorphic to a subgroups Aut(H).

c)If Z is the multiplicative group of complex numbers of absolute value 1 and Ri the additive group of reals then show that R/Z is isomorphic to T.

What happens if we replace R by Q?

6.a)Define the direct product of the subgroups H and k if and only if i)every x in G can be uniquely expressed as x = hk, h in H, k in K ii) hk = 1 if h in H, k in K.

b)Give an example of a group G and normal subgroups N1,N2,..,Nk such that G = N1N2...Nk and Ni intersection with Nj = {e} for all i is not equal to j and yet G is not direct product of Ni...Nk.

c)Prove that for n = kl, (k, l)=1

(z/nz) = (kz/nz) + (lz/nz)(direct).

7.a)Define p-subgroup of a group G (p is a prime).

Prove that for any prime p, every p-subgroup H of a finite group G is contained in a sylow p-subgroup of G.

b)Let p>q be primes and p is not congruent with 1 (modulo q). By using sylow's theorems show that every group of order pq is cyclic.

c)Determine all sylow subgroups of A4 , alternating group of degree 4.

8.a)Prove that a group G is soluble if and only if the derived series

G=G0containing G(1) containing G(2) containing ...of G ends at E in a finite number of steps.

b) If H and K are normal subgroups of G then show that G/(H intersection with K) is soluble if and only if G/H and G/K are both soluble.

c)i) Define a central series in a group G. Find the central series for a group Q8, group of quaternian.

ii) Is the group An(n>=4)nilpotent? why?

Book Name :

I.B.S.Passi and I.S.Luther

bamb_ssc@yahoo.com

http://mathskpo.freevar.com/N06.html

2. Originally Posted by bamb_ssc
1.a)If phi:G ->G' is a homomorphism of the group G into the group G' then with usual notations prove that

i)phi(1)=1

ii)phi(xn)=[phi(x)]n, x belongs to G, n belongs to Z

iii)phi preserves all group relations.
1) $\phi(ee) = \phi(e) \implies \phi(e) \phi(e) = \phi(e) \implies \phi(e) = e'$.
2) $\phi(x^n) = \phi(x\cdot ... \cdot x) = \phi(x)\cdot ... \cdot \phi(x) = \phi(x)^n$.
3)What is this supposed to mean?

b)If G is of finite order n then prove that the group of automorphisms of G (i.e.AutG) is isomorphic to the group of those positive integers <n, which are co-prime to n, with binary operation as multiplication modulo n and order of Aut(G)=phi(n)(phi is Euler's phi function).
This is not true just consider $G=S_6$ then $\text{Aut}(G) = S_6\times \mathbb{Z}_2$.

Aut(Zn,+n)is isomorphic to (Zncross,cross)
I assume you mean $\text{Aut}(\mathbb{Z}_n) \simeq \mathbb{Z}_n^{\times}$.
To prove this note if $\xi : \mathbb{Z}_n\to \mathbb{Z}_n$ is an isomorphism it is completely determined by $\xi ([1])$.
Now find the conditions to where $[1]$ can get mapped to.
It turn out it is $[1] \mapsto [k]$ where $\gcd(k,n)=1$ as the only possible ones.
Then it is easy to show that the automorphism group is $\mathbb{Z}_n^{\times}$

c)Construct an abelian group G of order 27 in which x3 = 1 for all x in G.
$\mathbb{Z}_3\times \mathbb{Z}_3\times \mathbb{Z}_3$

3. 2.a)i)Define a cycle of length l(i.e.l-cycle).

Prove that number of l-cycles in Sn is {(n)(n-1)...(n-l+1)}/l
Let $n=10$ and $l=8$.
Let $(X,X,X,X,X,X,X,X)$ be an $8$ cycle.
We will write the cycles with the smallest element first.
If $1$ is among these enteries we have $(1,X,X,X,X,X,X,X)$.
These leaves $7$ elements to be placed and here order is important - thus $\frac{9!}{2!}$ elements.
If $2$ is smallest then we have $(2,X,X,X,X,X,X,X)$.
These leaves $8$ elements to be placed - "1" was already used above.
Thus, we have $\frac{8!}{1!}$ elements.
If we have $3$ as smallest we have $(3,X,X,X,X,X,X,X)$ we have $\frac{7!}{0!}$ elements.
Thus, we have $\frac{9!}{2!}+\frac{8!}{1!}+\frac{7!}{0!}$ elements in total.

Now try to generalize this and prove the identity.

ii) If sigma is a cycle of length l, then prove that tsigmat-1 is also a cycle of length l for any t belongs to Sn.
I think you mean $\tau \sigma \tau^{-1}$.
Prove if $\sigma = (a_1a_2...a_k)$.
Then $\tau \sigma \tau^{-1} = (\tau(a_1)...\tau(a_k))$.

b)i) Define symmetric polynomial in n variables x1,x2,...xn.

Determine which of the following polynomial is symmetric polynomial

I)f(x, y, z)= x2y+y2z+z2x

II)g(x, y, z)= x2y + xy2 + y2z + yz2 + z2x + zx2.

ii)Find the group of symmetries of the polynomial f(x1,x2,x3)=product of(xi -xj), i<j.
Here

c)Determine the elements of order n in ccross, the group of non-zero complex numbers under multiplication.
Those are $z^n = 1 \implies z = e^{2\pi i k/n}$ for $k=0,1,...,n-1$.

4. 3.a)Define the cyclic structure of element sigma in Sn.

Prove that two elements of Sn are conjugate to each other if and only if they have the same cyclic structure.
We say two elements are similar iff they have the same number of disjoint cycles of equal length when factored into product of disjoint cycles. Thus, for example $(abc)(de)$ is similar to $(ABC)(DE)$ because they are expressed in disjoint cycles which are both two and they can be matched to have the same length.

To prove this we use the fact in 2ii.

b)Determine the conjugacy classes in S3 and A3 and hence write down their class equation.
The class equation for $A_3$ is $1+1+1$ since $|A_3|=3$ and any group of order three is abelian.
Now for $S_3$ we have $1$ for the identity element. We have $3$ for $(12),(13),(23)$.
We have $2$ for $(123),(132)$.
Thus, the class equation is $1+2+3$.

c)i)Show that in any group G, ab and ba are conjugate to each other. What can you say about the orders of ab and ba ?
Maybe you want it to be finite? Note Cauchy's theorem says we can embed any finite group into $S_n$. Thus, it remains to prove (or disprove) this for $S_n$.

ii)Rule out as many of the following as possible as class equations for a group of order 10.

1+1+1+2+5, 1+2+2+5, 1+2+3+4, 1+1+2+2+2+2.
The first one implies $|Z(G)| = 3$ and $3\not | 10$.

The second one is actually true, take $G=D_5$ - that is its class equation.

The third one means there is an element $a\in G$ with the number of elements in the conjugacy class of $a$ to be $4$. But then $4 = [G:C(a)]$ where $C(a)$ is the centralizer of $a$. This is impossible since $4\not | 10$ since $C(a)$ is a subgroup of $G$

The fourth one take $a,b,c,d$ to be the elements from the four different conjugacy classes of two elements. Then $C(a),C(b),C(c),C(d)$ all have index $2$ i.e. $|C(a)| = |C(b)| = |C(c)| = |C(d)| = 5$. Furthermore, each centralizer subgroup is different since they are different conjugacy classes. Since the orders of these groups are $5$ (a prime) it means the intersection of any two is trivial. And thus we have at least $5+4+4+4 > 10$ elements. A contradiction.

4.a)i)Show that a simple abelian group is cyclic of prime order.
This is very simple (pun).
It is not true for $G=\{e\}$ thus I assume $|G|>1$.
Choose $a\in G - \{ e\}$.
Now construct $\left< a \right>$ this is a subgroup.
Since it is abelian it must be a normal subgroup - thus $G = \left< a \right>$.
But now since $G$ is cyclic it means it has normal subgroups unless the order is prime.

ii)Show that for K=Z,Q,R or C, SL(n,K) is a normal subgroup of GL(n,K).
First $\mathbb{Z}$ is not a field.
Second $\text{SL}(n,K)\triangleleft \text{GL}(n,K)$.
This is because it is the kernel of the homomorphism $\det : \text{GL}(n,K) \to K^{\times}$.

b)State Cauchy's theorem for finite abelian group. Use this to prove that the converse of Lagrange's theorem for finite abelian group holds.
We can do better! Define a Dedekind group to be a group were all subgroups are normal. I found a generalization of this theorem to Dedekind groups here.

c)Show that a group of order 4 is either cyclic of isomorphic to V4.
This one is easy. Let $a\in V - \{ e\}$. Then $\text{ord}(a) = 2,4$. If each element in $V - \{ e \}$ has order two then it is Klein four otherwise if there is an element of order four then it is cyclic.

5. 5.a)If N,H are subgroups of a group G such that N is normal in G then prove that HN/N is isomorphic to H/(H intersection with N)
This is the 2nd Isomorphism Theorem.
Define $\phi : H \mapsto HN/N$ by $\phi (a) = aN$.
Now use the fundamental homomorphism theorem.
b)If N and C are respectively the normalizer and centralizer of a subgroup H of a group G, then show that C is normal in N and that N/C is isomorphic to a subgroups Aut(H).
This does not make sense, centralizer of what?
If you define the obvious $C(H) = \{ a\in G | aHa^{-1} = H\}$ then it makes no sense.
Because this is exactly what the normalizer is.

c)If Z is the multiplicative group of complex numbers of absolute value 1 and Ri the additive group of reals then show that R/Z is isomorphic to T.
This makes no sense because $Z\not \subseteq \mathbb{R}$.

6.a)Define the direct product of the subgroups H and k if and only if i)every x in G can be uniquely expressed as x = hk, h in H, k in K ii) hk = 1 if h in H, k in K.

b)Give an example of a group G and normal subgroups N1,N2,..,Nk such that G = N1N2...Nk and Ni intersection with Nj = {e} for all i is not equal to j and yet G is not direct product of Ni...Nk.
I think it has to be unique (at least in the finite case). Because say $G=H_1H_2$. Note $H_1H_2 = \{ab|a\in H_1,b\in H_2\}$. If this representation for an element was not unique i.e. $ab=a'b'$ then it would mean $|H_1H_2| < |H_1||H_2|$ since we know $|H_1\cap H_2||H_1H_2| = |H_1||H_2|$. And I think this argument generalizes.

c)Prove that for n = kl, (k, l)=1

(z/nz) = (kz/nz) + (lz/nz)(direct).
Just use the definition you provided above and the fact that any $a\in \mathbb{Z}$ can be written as $a=kx+ly$ since $\gcd(k,l)=1$.

7.a)Define p-subgroup of a group G (p is a prime).
A group where the order of any element is a power of a prime.

Prove that for any prime p, every p-subgroup H of a finite group G is contained in a sylow p-subgroup of G.
This is Sylow's 2nd Theorem.

b)Let p>q be primes and p is not congruent with 1 (modulo q). By using sylow's theorems show that every group of order pq is cyclic.
The number of Sylow $p$-subgroups is unique. And the number of Sylow $q$-subgroups is unique. Their intersection must be zero. And thus, $G$ is isomorphism to $C_p\times C_q \simeq C_{pq}$ which is cyclic.

c)Determine all sylow subgroups of A4 , alternating group of degree 4.
It has one Sylow subgroup of order four which is $\{ \text{id},(12)(34),(13)(24),(14)(23)\}$.
It has four Sylow subgroups of order three.
I give you one of them you can find the other three, $\left< (123) \right>$.

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# class equation of the group of order 10

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