Articles of group theory

Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by $(e^{2i1\pi/n}, e^{2i2\pi/n},\dots, e^{2ik\pi/n},\dots,e^{2in\pi/n}, 0, 0, 0,\dots$). Now consider the set S of all such sequences, $$S=\left\{(e^{2i1\pi/n}, e^{2i2\pi/n},\dots, e^{2ik\pi/n},\dots,e^{2in\pi/n}, 0, 0, 0,\dots) : […]

Prove or disprove that $\exists a,b,c\in\mathbb{Z}_+\ \forall n\in\mathbb{Z}_+\ \exists k\in\mathbb{Z}_+\colon\ a^k+b^k+c^k = 0(\mathrm{mod}\ 2^n)$

Prove or disprove that there exist a triplet of positive integers $(a,b,c)$ with $\mathrm{gcd}(a,b,c)=1$ such that for any positive integer $n$ there exist such positive integer $k$ that $$ a^k+b^k+c^k = 0(\mathrm{mod}\ 2^n) $$ First thing that I’ve noticed is that one and only one of the numbers $a,b,c$ is even, otherwise either all of […]

how many element of order 2 and 5 are there

$G=Z_{10}\times Z_{15}$ Then $G$ contains exactly one element of order $2$ five element of order $3$ 24 elements of order $5$ 24 elements of order $10$ what I think is 1 is correct, not able to guess others.

$G$ is $p$-supersoluble iff $|G : M | = p$ for each maximal subgroup $M$ in $G$ with $p | |G : M |$.

Let $G$ be a $p$-soluble group. Then $G$ is $p$-supersoluble if and only if $|G : M | = p$ for each maximal subgroup $M$ in $G$ with $p | |G : M |$.

Is there an example of a non-orientable group manifold?

Basically what I’m looking for is a topological group that is also a non-orientable, n-dimensional manifold

Proving a set is an abelian group.

I am trying to prove that $(G, *)$ is an abelian group with $G=(-1,1)$ and $a*b=$$\frac{x+y}{1+xy}$. Thus far I have found that the identity element $e=0$. From here, I set $a*b=0$ and found $a^{-1}$ to be $-a$. My work for trying to prove closure and that the set is abelian is: Let $a,b \in G$, […]

How far can we go with group isomorphisms?

The following is quoted from Wikipedia: From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. But this statement is too general. What I’m wondering about is whether there is a limit to these properties for which isomorphic groups “need not be distinguished”. For instance, I have been […]

Intuition – $fr = r^{-1}f$ for Dihedral Groups – Carter p. 75

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square’s vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles’s $f$ is different. Carter fleshes out why $frf = r^{-1} $ intuitively: (1.) Can someone please unfold, like Carter, why $fr = r^{-1}f $? I see why for this […]

Group Order and Least Common Multiple

Let $G_1,G_2,…,G_n$ be groups. Show that the order of an element $(a_1,a_2,…,a_n)$ $\in$ $G_1 \times G_2 \times\cdots\times G_n$ is lcm($o(a_1),…,o(a_n))$. I know I need to use the fact that the least common multiple of positive integers $x_1,x_2,…,x_n$ is the unique positive multiple of $x_1,x_2,…,x_n$ that divides all other such multiples. Note on notation: for the […]

The centralizer of an element x in free group is cyclic

If $F$ is free group and $1 \neq x \in F$, then $C_F(x)$ is cyclic. help me please!