Tagged Questions

1
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2answers
87 views

Can you find an isomorphic group?

Let $G$ be a group with elements $\{e, a, b, c, \theta, \theta a, \theta b, \theta c \}$ where $a^2 = b^2 = c^2 = \theta$, $\theta^2 = e$, $ab = \theta b a = c$, $bc = \theta c b = a$, $ca = \theta a ...
2
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2answers
126 views

Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$} Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a ...
2
votes
1answer
54 views

Proving that an element in an algebra presentation is nonzero

Let $F$ be a field, $F\langle x,y\rangle$ the free $F$-algebra on two generators (polynomials in two noncommuting variables), $A= F\langle x,y\,|\, xy\!=\!1\rangle= F\langle x,y\rangle/\langle\langle ...
1
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1answer
80 views

Solving conjugacy equations in dihedral groups.

For all integers $m$ such that $0≤m<n$ find $a,b,c\in D_n$ such that $a(rR^m ) a^{-1}=R^2$ $b(rR^m ) b^{-1}=r$ $c(rR^m ) c^{-1}=rR$ $D_n$ is dihedral group of an $n$-gon represented by ...
1
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3answers
133 views

How do I find $\left|\langle a,b\mid a^2=b^3=e\rangle\right|$?

Suppose $G$ is a group satisfying $G=\langle a,b\mid a^2=b^3=e\rangle$. Find $|G|$.
3
votes
3answers
62 views

Does there exist a finite group with the following presentation?

Let $G$ be a finite group (with only two generators and $m=n$) presented as $$ G = \langle a, b : a^m = b^n = (W(a,b))^p= \ldots\text{other-such-relations}\ldots= 1 \rangle $$ where $m,n,p>1$ , ...
2
votes
1answer
63 views

What are some slick ways to prove that a presentation is actually isomorphic to a given group?

Let's say I have a particular finite presentation and want to show it's actually a presentation for the group I claim it's a presentation for. That group might be specified, say, by a linear or ...
0
votes
1answer
89 views

If $a^{3^3}=b^9=1$ for generators $a,b$ of $G$, can we conclude that $G$ is a $3$-group?

I know that $a$ and $b$ are generators of a group $G$ and $a^{3^3}=b^9=1$. Are these informations sufficient to affirm that the group is a $3$-group? Adding the relation $b^{-1}ab=a^4$, can we ...
1
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2answers
163 views

$\langle X|\emptyset\rangle\ncong\langle X|R\rangle$ for finitely presented groups (exercise in Massey)

Let $:F_n$ denote the free group of rank $n$. How can I solve the exercise 7.6.3.(b), page 234, in Massey's Algebraic Topology? I'm guessing there's been made a mistake and (b) actually reads "If ...
2
votes
1answer
147 views

Quotient of free group with normal subgroup

I'm trying to make up an example of a quotient of a free group to check if I understand quotients properly. I do for the usual cases but I've not seen free groups before. So here I go: Let $F = ...
2
votes
2answers
216 views

free groups: $F_X\cong F_Y\Rightarrow|X|=|Y|$

I'm reading Grillet's Abstract Algebra. Let $F_X$ denote the free group on the set $X$. I noticed on wiki the claim $$F_X\cong\!\!F_Y\Leftrightarrow|X|=|Y|.$$ How can I prove the right implication ...
0
votes
1answer
134 views

the formulation of Nielsen-Schreier theorem: every subgroup of a free group is free

I just got a little confused reading the formulation on wiki. Let $F_X$ denote the free group on the set $X$ and let the symbol $\leq$ denote "is subgroup of". From what I know, the theorem reads: ...