1
vote
2answers
40 views
Suppose a finite group $G$ has an element $a$ which…
I was solving past exam papers and was stuck on the following one:
Suppose a finite group $G$ has an element $a$ which is not the identity such that $a^{20}$
is the identity. Which of the following ...
3
votes
0answers
34 views
Discrete subgroups of SU(n) and SO(n).
Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful.
I would like to know whether there is a complete understanding of discrete ...
-1
votes
2answers
46 views
General Linear Groups with Homomorphisms [closed]
Let $G=\mathrm{GL}_n(\mathbb R)$ and $H=\mathbb R^*$. Let $\phi : G=\mathrm{GL}_n(\mathbb R) \rightarrow \mathbb R^*$ be the map defined by $\phi(A)=\det(A)$. Show that $\phi$ is a group ...
6
votes
0answers
50 views
Ulm and Frattini Subgroups
Let $A$ be an abelian group. We define $U(A)=\cap (nA), n\in \mathbb N$ be the Ulm subgroup of $A$. The Frattini subgroup of $A$ is $\Phi(A)=\cap(pA)$ ($p\in \mathbb P$). I was trying to show that ...
2
votes
1answer
31 views
Group extension of $\mathbb Z_4$ by $\mathbb Z_2$
Let $f : G →\mathbb Z_2$ be an extension of $\mathbb Z_4$ by $\mathbb Z_2$. Suppose that the induced action $α_f :\mathbb Z_2 →\mathbb Z^{\times}_4$ carries the generator of $\mathbb Z_2$ to $−1$. ...
6
votes
0answers
78 views
Orders of elements and homomorphisms.
Corollary 4.6.8 There is a group $G$ of order $n^3$ given by $G= \{b^ic^ja^k \mid 0 ≤ i, j, k < n\}$, where $a$, $b$, and $c$ all have order $n$, and $b$ commutes with $c$, $a$ commutes with ...
2
votes
1answer
51 views
Proving that $X$ is a subgroup of $G$
If we're given that some $X\subset G $ such that $e\in X$, and $\forall g \in G$, cosets $gX$ partition $G$, is $X$ a subgroup of $G$?
I'm not quite sure what it is I have to do. If $X$ wasn't a ...
3
votes
2answers
51 views
Find $\mathrm{Aut}(G)$, $\mathrm{Inn}(G)$ and $\mathrm{Aut}(G)/\mathrm{Inn}(G)$ for $G = D_4$
Problem
Find $\mathrm{Aut}(G)$, $\mathrm{Inn}(G)$ and $\mathrm{Aut}(G)/\mathrm{Inn}(G)$ for $G = D_4$
My Attempt
I let $D_4 = \{e, x, y, y^2, y^3, xy, xy^2, xy^3\}$
I found that $\mathrm{Inn}(G)$ ...
2
votes
2answers
90 views
+50
All of the dihedral groups are factor groups of the infinite dihedral group.
Show that $\operatorname{Aut}(\Bbb{Z}) \cong \{\pm 1 \}$ and write $\alpha : \mathbb Z_2 \rightarrow \operatorname{Aut}(\Bbb{Z})$ for the nontrivial homomorphism. The semidirect product $\Bbb{Z} ...
3
votes
1answer
41 views
Reference request for ordered groups
I've been reading Pete Clark's notes on commutative algebra, and I especially liked section 17 on valuation rings, and ordered groups in particular.
I'm looking for more introductory material ...
0
votes
2answers
53 views
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$.
Here is what I have here:
Aut$(G)$ consists of 6 bijective functions, which maps $G$ to ...
4
votes
3answers
66 views
Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.
Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at
least two elements.
We know that Aut($G$) contains the identity function $f: G \to G: x \mapsto x$.
If $G$ is ...
0
votes
1answer
87 views
Is $\{a + bi: a, b \in \mathbb{F}_3\}$ a field?
Is $P = \{a + bi: a, b \in \mathbb{F}_3\}$ a field? This is the question I am having.
First, I list the elements in $\mathbb{F}_3$, which consists of $\overline{0}$, $\overline{1}$ and ...
5
votes
1answer
69 views
Order of elements in a group.
Corollary 4.6.8. There is a group $G$ of order $n^3$ given by $G = \{b^ic^ja^k | 0 ≤ i, j, k < n\}$, where $a$, $b$, and $c$ all have order $n$, and $b$ commutes with $c$, $a$ commutes with $c$, ...
2
votes
1answer
35 views
Minimal Polynomials Annihilating an Abelian Torsion-Free Group
Let $A$ be an abelian torsion-free group. Let $\theta \in\operatorname{Aut}A$. Assume that $\theta$ has a finite period in $\operatorname{Aut} A$, say $n$. Obviously $\theta^n-1$ annihilates $A$ (i.e. ...
5
votes
0answers
47 views
Why are the p-adic integers a linearly ordered group? [duplicate]
In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group.
I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
4
votes
2answers
93 views
Finite abelian $2$-group
If $G$ is a finite abelian group such that $o(x)=2$ for all $x \neq e$ and $|G|=2^n$ for some $n\in\mathbb N$, prove that $G \cong \mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ ($n$ factors).
Any ...
8
votes
2answers
35 views
The index of $\xi_4^*$ in $\xi_4$
Just seeing if i'm right:
With the set of solutions for $z^4=1$: $\xi_4=\{1,i,-1,-i\}$, one can construct the group of the $4$th roots of unity: $(\xi_4,\cdot_\mathbb{C})$ and its multiplicative ...
3
votes
1answer
28 views
Show that the subgroup $s(K) \subset H \rtimes_{\alpha} K$ is normal if and only if $\alpha: K \rightarrow Aut(H)$ is the trivial homomorphism.
Show that the subgroup $s(K) \subset H \rtimes_{\alpha} K$ is normal if and only if $\alpha: K \rightarrow Aut(H)$ is the trivial homomorphism, where $s : K \rightarrow H \rtimes_{\alpha} K$ is given ...
0
votes
2answers
52 views
Rings | Homomorphisms | Units
Question
Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is
a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$.
Attempt
...
6
votes
1answer
44 views
Given $G$, when can we find a division ring $R$ with $R^*=G$?
This is motivated by a characterization of finite cyclic groups, in which one proves
Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic.
The proof is ...
2
votes
2answers
57 views
question on subgroups of prime order
Let $G$ be a group and let $\,H,\, K\,$ be subgroups of $\,G,\,$ each of order $\,p,\,$ where $\,p\,$ is prime.
Show that either $\,H\cap K =\{e\},\,$ or $\,H=K.\,$
Is the result true if ...
1
vote
3answers
87 views
Prove $rs=sr^{-1}$ in ${\rm Dih}(2n)$
Let $r$ and $s$ be the rotation and reflection symmetries respectively in ${\rm Dih}(2n)$, the dihedral group of order $2n$. Show that $rs=sr^{-1}$.
I also need to show by induction that ...
3
votes
0answers
50 views
Can all non-archimedean groups be written as a product of archimedean groups?
All the non-archimedean groups I know of can be written as the product of archimedean groups. I'm wondering if this is generally true. I think I've found a proof, but I haven't heard this theorem ...
1
vote
2answers
46 views
The set of complex numbers of modulus $1$ is a group under multiplication
Show that $C=\{z\in \mathbb{C} \mid |z|=1\}$ is a group under complex multiplication.
I'm a little confused because isn't the identity the only element with order $1$? What is this set?
4
votes
2answers
61 views
Cardinality of $GL_n(K)$ when $K$ is finite
I don't know how to do the last task of an exercise.
Let $K$ be a field, $G=GL_n(K)$ and $X=K^n\backslash\{0\}$.
First task: Show that $G \times X \to X$, $(A,x)\mapsto Ax$ defines an action of $G$ ...
4
votes
3answers
80 views
$\mathbb{Q}/\mathbb{Z}$ has cyclic subgroup of every positive integer $n$? [duplicate]
I would like to know whether $(\mathbb{Q}/\mathbb{Z},+)$ has
$1$. Cyclic subgroup of every positive integer $n$?
$2$. Yes, unique one.
$3$. Yes, but not necessarily unique one.
$4$. Does not have ...
5
votes
0answers
46 views
+100
Amenable group rings embeddable in skew fields
I'm looking for a reference of the following fact:
given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent:
(1) the group ring $K[G]$ is a domain;
(2) $K[G]$ is ...
5
votes
3answers
97 views
Homomorphism from $\mathbb{Z}/n\mathbb{Z}$
Does there exist a homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly?
4
votes
3answers
63 views
Homomorphisms between $ \mathbb{Z} $ modules.
Calculate $\newcommand\Hom{\operatorname{Hom}}\Hom(\mathbb Z \oplus \mathbb Z_{p^\infty},\mathbb Z \oplus \mathbb Z_{p^\infty})$. Where $ \mathbb Z_{p^\infty}= ...
2
votes
0answers
22 views
Extending transvections/generating the symplectic group
The context is showing that the symplectic group is generated by symplectic transvections.
At the very bottom of http://www-math.mit.edu/~dav/sympgen.pdf it is stated that any transvection on the ...
4
votes
3answers
77 views
on the commutator subgroup of a special group
Let $G'$ be the commutator subgroup of a group $G$ and $G^*=\langle g^{-1}\alpha(g)\mid g\in G, \alpha\in Aut(G)\rangle$.
We know that always $G'\leq G^*$.
It is clear that if $Inn(G)=Aut(G)$, then ...
2
votes
2answers
48 views
Direct Product of the $G_i $'s
I am a little confused in the interpretation of the product of groups. Here is what's written in my notes:
Given groups $G_1,G_2,...,G_n$, recall that $G_1\times G_2\times ...\times ...
2
votes
3answers
58 views
How to find all elements of $\mathbb{Z}_{4} \times\mathbb{Z}_{4}/\langle(1,1)\rangle$?
I am studying factor groups, and I saw an example that says
Find all the elements of the factor group $\mathbb{Z}_{4} \times \mathbb{Z}_{4}/\langle(1,1)\rangle$.
I know that the order of ...
4
votes
2answers
64 views
Which one of the following groups is decomposable?
A group $(G,+)$ is said to be decomposable if $G$ has two non-trivial subgroups $G_1$ and $G_2$ such that $G=G_1+G_2$ and $G_1 \cap G_2 =$ {$e$}. Then which of the following are decomposable:
(i) ...
2
votes
3answers
87 views
Why can't this be a coset?
Let $H$ be a subgroup of $G$ and H is not normal, there are left cosets $aH$ and $bH$ whose product isn't a coset.
My attempt:
$ab H\subset aHbH$ and if H is not normal, if $ah_1bh_2=abh_3$ for all ...
2
votes
2answers
48 views
Rotman's Introduction to to the theory of groups. Exercise 3.45.
Can you give me a hint on the first part of the exercise?
Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
2
votes
1answer
83 views
What does it mean to “Decide to which group $G$ is isomorphic” for a given group $G$?
I have a homework question which is
Decide to which group $(\mathbb{Z}_n^*,\,\cdot\,)$ is isomorphic (classification of finite abelian groups), for
(i) $n = 9$,
(ii) ...
3
votes
2answers
62 views
Silliness: $\exists~X~\text{s.t.}~AX=B \iff B\in R(L_A)$
So, I am asked to prove that the system of linear equations $AX=B$ has $\color{black}{a~solution}$ if and only if $B\in R(L_A)$. $R$ denotes the "range of" and $L_A$ is left multiplication by $A$. If ...
7
votes
1answer
71 views
Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that the order of the center of $G$ is 1 or $pq$.
Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that
the order of the center of $G$ is 1 or $pq$.
Let me start off with what I did:
Assume $G$ is abelian. Then we know ...
1
vote
1answer
35 views
$P \cap N$ is a Sylow $p$-subgroup of $N$, where $N$ is normal in $G$ and $P$ is a Sylow $p$-subgroup of $G$?
In 'A Course in Group Theory' by Humphreys, Proposition 11.14 says that if $G$ is a finite group, $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$, then $P \cap N$ is a Sylow ...
5
votes
3answers
100 views
What is the meaning of the parentheses in $\phi^{-1}\left[\{\phi(g)\}\right]=gH=Hg$?
I am studying homomorphisms is groups and i saw a theorem saying:
For $g$ in a group $G$, the cosets $gH$ and $Hg$ are the same, and collapsed onto the single element $\phi(g)$ by $\phi$. That is, ...
7
votes
1answer
63 views
Why does the automorphism used to construct the group have to be non-inner?
I have a question on why a particular assumption is made that the automorphism used to construct a certain group be non-inner.
In [Herstein, Topics in Algebra, p. 69], a construction of a nonabelian ...
-1
votes
1answer
60 views
Generators of a cyclic group and their orders
a) Let $G = \langle a \rangle$ be a finite cyclic group. Prove that for each $b\in G$, $\langle b \rangle=G$ if and only if order of $b$ equals order of $G$.
b) The previous part does not hold if $G$ ...
3
votes
0answers
71 views
How to recover the integral group ring?
I would like to solve the following exercise:
Suppose $R$ is a commutative semisimple ring of characteristic $p^t, t\geq1$, and we have two finite groups $G_1=H_1 \times A_1$ and $G_2=H_2 \times ...
10
votes
0answers
72 views
Show that $h \equiv 1 \pmod p$, where $h$ is the number of subgroups of order $p$ and $p$ divides the group order. [duplicate]
Let $G$ be a finite group and $p$ a prime number that divides the
order of $G$. Let $h$ be the number of subgroups of $G$ of order $p$.
Prove that there are $h(p-1)$ elements of order $p$ in ...
3
votes
3answers
51 views
Quotient groups and homomorphism
If $G$ and $H$ are groups. Let $G^\star= \{(a, e_H)| a\in G\}$ and $H^\star=\{(e_H, b) |b \in H\}$. Show that
$(G \times H)/G^\star$ is isomorphic to $H$ and $(G \times H)/H^\star$ is isomomorphic to ...
3
votes
0answers
38 views
reflection groups and hyperplane arrangement
We know that for the braid arrangement $A_\ell$ in $\mathbb{C}^\ell$: $$\Pi_{1 \leq i < j \leq \ell} (x_i - x_j)=0,$$
$\pi_1(\mathbb{C}^\ell - A_\ell) \cong PB_\ell$, where $PB_\ell$ is the pure ...
3
votes
1answer
48 views
How many elements of order $7$ are there in a group of order $28$ without Sylow's theorem
How many elements of order $7$ are there in a group of order $28$
I need to prove this result without using the Sylow's Theorem.By Sylow's Theorem it has only one subgroup and the anser becomes ...
9
votes
1answer
188 views
What can we say about the size of $HK\cap KH$ when $HK\neq KH$?
If $G$ is a finite group, and $H$, $K$ are proper subgroups of $G$, then it is not necessary that $HK=KH$. But, these two subsets have same size. The question I would like to ask, then, is
If ...
