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 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 […]

$G=Z_{10}\times Z_{15}$ 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.

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 $p | |G : M |$.

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

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 $a,b \in G$, […]

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 […]

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 […]

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 […]

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

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