The study of symmetry: groups, subgroups, homomorphisms, actions
1
vote
2answers
46 views
$G$ $p$-group. If $H\triangleleft G$, then $H\cap C(G)\ne \{e\}$
I'm pretty new on this subject and I need a hint to begin to solve this question:
If $G$ is a finite p-group, $H\triangleleft G$ and $H\ne \{e\}$, then
$H\cap C(G)\ne \{e\}$
Thanks for any ...
1
vote
3answers
26 views
Subgroups of cyclic groups with same order
Let $C$ be a cyclic group. Let $A$ and $B$ be two subgroups of $C$ with $|A|=|B|$. Then $A = B$.
How to show this? Thanks.
(Btw, I already know $A$ and $B$ are cyclic)
5
votes
3answers
104 views
If $G$ has only 2 proper, non-trivial subgroups then $G$ is cyclic
Is the following true?
If $G$ has two proper, non-trivial subgroups then $G$ is cyclic.
-2
votes
1answer
154 views
Help with abstract algebra
Let $G=\{1,-1,i,-i,j,-j,k,-k\}$ where $i^2 =j^2 =k^2 =-1$, $-i=(-1)i,$ $1^2 =(-1)^2 =1$, $ij=-ji=k$, $jk=-kj=i$, and $ki=-ik=j$.
a) Construct the Cayley table for $G$
b) Show that $H=\{-1,1\}$ is ...
3
votes
2answers
46 views
Show $(\mathbb{C}^*,\cdot,1) \cong (\mathbf{T},\cdot,1)\times (\mathbb{R}_{>0},\cdot,1)$ where $\mathbf{T} = \{a+bi \in \mathbb{C}|a^2+b^2=1\}$.
Show $(\mathbb{C}^*,\cdot,1) \cong (\mathbf{T},\cdot,1)\times (\mathbb{R}_{>0},\cdot,1)$ where $\mathbf{T} = \{a+bi \in \mathbb{C}|a^2+b^2=1\}$ and $\mathbb{C}^*=\mathbb{C}\setminus \{0\}$.
So ...
3
votes
1answer
58 views
The order of a conjugacy class is bounded by the index of the center
If the center of a group $G$ is of index $n$, prove that every conjugacy class has at most $n$ elements. (This question is from Dummit and Foote, page 130, 3rd edition.)
Here is my attempt: we have
...
2
votes
1answer
282 views
Find a subgroup of the octic group that is normal, and one that is not normal
Given the octic group $G = \{e, \sigma, \sigma^2, \sigma^3, \beta, \gamma, \delta, t\}$.
Find a subgroup of $G$ that has order $2$ and is a normal subgroup of $G$.
Find a subgroup of $G$ that has ...
1
vote
1answer
55 views
Showing that a specific function $\mathbb Z[x]\to\mathbb Z$ is a group homomorphism.
Let $\mathbb Z[x]$ be the group of polynomials in an indeterminate $x$ with integer coefficients under addition. Prove that mapping from $\mathbb Z[x]$ into the group $\mathbb Z$ given by mapping ...
4
votes
2answers
52 views
Finite Groups: $a \in G \implies a \in H$
Let $G$ be a finite group and let $H$ be a normal subgroup. Let $a$ be an element
of G and suppose that $\gcd(|a|,[G : H]) = 1$. Show that $a$ is in $H$.
3
votes
1answer
24 views
Primitve roots and congruences?
Let $p$ be an odd prime. Show that the congruence $x^4$$\equiv -1\text{ (mod }p\text{)}\
$ has a solution if and only if $p$ is of the form $8k+1$.
Here is what I did
Suppose that $x^4$$\equiv ...
4
votes
0answers
58 views
Groups acting on polytopes
I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs.
Their basic ...
0
votes
1answer
48 views
Let $G$ and $H$ be groups, and let $X:G\to H$ be a group homomorphism.
Let $G$ and $H$ be groups, and let $X : G\to H$ be a group homomorphism. Prove the
following statements.
A. If $K$ is a subgroup of $H$, then $X^{-1}
(K)$ is a subgroup of $G$.
B. If $K$ is normal ...
1
vote
2answers
57 views
Isomorphisms of Product Groups
If $G \cong H \times \mathbb{Z}_2, $ show that $G$ contains an element $a$ of order $2$ with the property that $ag = ga$ for all $g \in G$. Deduce (briefly!) that the dihedral group $D_{2n+1}$ (with ...
6
votes
5answers
321 views
Is $\mathbb{Z}^2$ cyclic?
Is $\mathbb{Z}^2$ cyclic? What does it means for a group to be cyclic? Is it just that it has one generator?
Thanks
1
vote
1answer
55 views
Finding a primitive root modulo $11^2$
Find a primitive root modulo each of the following moduli:
a) $11^2$
My TA said he is not going to go over this so do not worry about it. He said you can try this if you want but he would ...
