Tagged Questions

0
votes
1answer
51 views

How does one prove U(100) is not cyclic without computing the order of any element?

For starters, $|U(100)| = \varphi(100) = 40$. So by Lagrange, $\forall \hat{x} \in U(100), ord(\hat{x}) \mid 40$. Now, how can one prove there aren't any generators in U(100) without computing the ...
4
votes
1answer
50 views

Group homomorphisms into a field

Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: ...
0
votes
2answers
61 views

How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^\star \to (\mathbb{Z}/154\mathbb{Z})^\star $ where $f(x)=x^5$?

How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^* \to (\mathbb{Z}/154\mathbb{Z})^* $ where $f(x)=x^5$? The group operation in this case is multiplication with ...
2
votes
0answers
37 views

Find the cycle index of cyclic group $C_p$ where $p$ is prime

Find the cycle index of cyclic group $C_p$ where $p$ is prime. No idea where to start...any help pointing me in the right direction would be appreciated.
1
vote
3answers
76 views

Why is $| \rm{Aut}(\mathbb{Z}_n) | = \phi(n)$?

If $G$ is a finite cyclic group of order $n$, prove that $| \rm{Aut}(G) | = \phi(n)$, where $\phi(n)$ is the Euler's totient function. Can someone please help me with this?
0
votes
3answers
66 views

Groups - Prove that if $G/Z(G)$ is cyclic then $G$ is abelian

Prove that if $G/Z(G)$ is cyclic then $G$ is abelian. Using this fact and $G$ is a nontrivial group of prime power order, deduce that a group of order $p^2$ , $p$ prime, is abelian.
1
vote
1answer
65 views

Euler's formula and subgroups of $\mathbb Z_n$

Prove that in $\mathbb Z/n\mathbb Z$ for every divisor $d$ of $n$ there is a unique subgroup of order $d$ using the following results: $\sum_{d\mid n}\varphi(d)=n$ and the number of generators of ...
1
vote
2answers
68 views

Orders of elements in cyclic groups

I think I'm a bit confused about the order of elements in cyclic groups. If we suppose $G$ is a group of order 35, and let $x∈G$ such that x≠e, from Lagrange's Theorem $x$ will be of order 5, 7, ...
1
vote
2answers
94 views

Sylow Theorems Question

Let $p$ be the smallest prime that divides the order of a finite group $G$. If $H$ is a Sylow $p$-subgroup of $G$ and is cyclic. Prove that $N(H)=C(H)$.
3
votes
5answers
136 views

What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$

What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$ and how to find it? I came across the above problem and do not know how to get it? Can someone point me ...
1
vote
2answers
77 views

Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .

I'm looking for a simple proof that up to isomorphism every group of order 2p (p prime) is either $\mathbb{Z}_{2p}$ or $D_{p}$ (The Dihedral group of order 2p). I should note that by simple I mean ...
2
votes
2answers
97 views

How do I find the elements of $S=\langle (12),(1324)\rangle$?

Let $G=S_4$, I want to find the elements of $$S=\langle (12),(1324)\rangle$$ How do I do this? I know that $$\langle(1324)\rangle≤S$$and I know I have to find out orders of $(12)$ and $(1324)$ ...
1
vote
4answers
136 views

group generator

Need to prove this : Let $a$ element in $\mathbb Z_n$ , then $a$ is the generator of group $(\mathbb Z_n,+)$ iff $\gcd(a,n)=1$ Have no idea how to prove this, would appreciate some guidance.
2
votes
1answer
49 views

The order of a subgroup of a cyclic subgroup

Let $G = Z^*_p$ under multiplication, with $p$ being a prime $> 3$ and $|G|=p-1$ and $G$ is cyclic. $H = \{a^2\mid a \in G\}$. Want to prove $H < G$ and $|H| = (p-1)/2$. I can show that $H < ...
1
vote
2answers
60 views

Computing the order and cyclicity of quotients of direct products.

Determine the order of $(\mathbb{Z} \times \mathbb{Z} )/ \langle(2,2)\rangle$ and $(\mathbb{Z} \times \mathbb{Z} )/ \langle(4,2)\rangle$. Are the groups cyclic? I've read many solutions on the ...
2
votes
2answers
52 views

Checking that $\rho$ is a representation

Let $z$ be a generator of the cyclic group $\mathbb{Z}_3 = \{ 1,z,z^2 \}$. Prove that a representation $\rho$ of $\mathbb{Z}_3$ in the $2$-dimensional complex vector space $\mathbb{C}^2$ can be ...
1
vote
2answers
75 views

Generators of a finite additive cyclic group

Let $C$ be an abelian additive group and write e for a generator of $C$. The elements of $C$ are then $0,e,2e,3e,\dots,(n-1)e$. If $C$ is finite, prove that the element $ke$ is another generator of ...
2
votes
2answers
247 views

Prove that there are no simple groups of order 224.

Prove that there are no simple groups of order 224. Let $G$ be a finite group such that $\vert G \vert = 224 = 2^5 \cdot 7$. We know that $n_2 \mid 7$ and $n_2 \equiv 1 \pmod 2$ and we know that $n_7 ...
2
votes
1answer
95 views

$G$ finite group, $H \trianglelefteq G$, $\vert H \vert = p$ prime, show $G = HC_G(a)$ $a \in H$

Let $G$ be a finite group. $H \trianglelefteq G$ with $\vert H \vert = p$ the smallest prime dividing $\vert G \vert$. Show $G = HC_G(a)$ with $e \neq a \in H$. $C_G(a)$ is the Centralizer of $a$ in ...
2
votes
2answers
134 views

Necessary and sufficient condition for one cyclic group to be the subset of some other cyclic group.

My professor gave us this problem, wondering if anyone could help me out: Suppose a is an element of order n in a group G. Find a necessary and sufficient condition for which $\langle a^r\rangle ...
1
vote
2answers
99 views

Cycling through powers of a generator of finite field. Something similar to modpow for Z/nZ

I am not sure if I am talking to the correct community (perhaps stack overflow is best). I want to be able to compute $$g^x = \underbrace{g \cdot g \cdot \dots \cdot g}_\text{x amount of times}$$ ...
1
vote
3answers
106 views

Order of a subgroup of a finite cyclic group

Let $G$ be a cyclic subgroup of order $n$, generated by say $a\in G$ where the identity of $G$ is labelled $e$. Let $H$ be the cyclic subgroup of $G$ generated by some $a^{m}\in G$. Then I want to ...
1
vote
1answer
65 views

What are the generators for $\mathbb{Z}_p^*$ with p a safe prime?

lets consider $\mathbb{Z}_p^*$ with $p = 2 \cdot q + 1$ a safe prime ($p$ and $q$ have to be prime). Then $\varphi\left(p\right) = 2 \cdot q$ is the order of $\mathbb{Z}_p^*$, and ...
3
votes
4answers
803 views

Homomorphism between cyclic groups

I have some confusion in relation to the following question. Let $\langle x\rangle = G$, $\langle y\rangle = H$ be finite cyclic groups of order $n$ and $m$ respectively. Let $f:G \mapsto H$ ...
3
votes
3answers
222 views

Is there a simple way to distinguish between group homomorphisms?

More precisely, I am given a function $f:G\to H$ with the promise that it is a homomorphism. Is there an easy way to determine which homomorphism it is without looking through all of its values? For ...