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 Yearling
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Aug
10
comment Can I use induction by $|V|$ here?
It's probably easier to use induction on the number of edges, and look at what happens when you delete an edge.
Aug
4
awarded  Yearling
Jul
16
answered Definition of quasi-cyclic and full rational groups
May
8
answered Suppose $G = G_1 * G_2$. let $c \in G$ and let $A = cG_1c^{-1}$. Show that $A\cap G_2 = \{1\}$
May
7
comment Abelianised dihedral group isomorphism
Suppose that $n =2k$. You have $x^2 = 1$ and $x^{2k}=1$. But $x^{2k}=1$ follows from $x^2=1$ because $x^{2k} = (x^2)^k$.
May
7
answered Abelianised dihedral group isomorphism
May
1
comment Implications of the order of subgroups, given the order of the parent is a composite number
Yes, if the $x_i$ are all primes; this is essentially Cauchy's theorem.
Apr
30
comment “Quotient” as a verb
People are just verbing the word "quotient".
Apr
29
answered Finitely presentated subgroups of a group are normal?
Apr
29
comment Distinct four elements in G
It's not true. The cyclic group of order $5$ has exactly four (non-trivial) elements whose fifth power is the identity, and they are certainly distinct.
Apr
29
comment What is the easiest way to generate $\mathrm{GL}(n,\mathbb Z)$?
@Leon. Diagonal matrices are elementary matrices.
Apr
23
comment Generate specific reduced words that “violate freeness”
I'm not sure I understand the problem. What hypothesis do you think you need about the order of $g_1$? If everything collapses, I think you're going to contradict your assumption that $g_1$ and $g_2$ do not commute.
Apr
23
comment Generate specific reduced words that “violate freeness”
Yes, for (1), you can conjugate the word by $g_{i(1)}^{m(1)}$ to get the first and last syllables to be distinct. I don't yet understand the second question, though.
Apr
23
comment Trying to show $|ab|$ divides lcm$(|a|,|b|)$
Try looking at the set of distinct (cyclic) subgroups of order $p$.
Apr
23
comment Paradoxical Decomposition
@AllAboutGroups We can write $F_2 = B\mathscr{F}(B^{-1})\cup\mathscr{F}(B)$, and $F_2 = A\mathscr{F}(A^{-1})\cup\mathscr{F}(A)$ where, for $X\in\{A,B,A^{-1},B^{-1}\}$, we define $\mathscr{F}(X)$ to be the set of reduced words in $F_2$ beginning with $X$.
Apr
20
answered Paradoxical Decomposition
Apr
20
comment Is $A_4$ isomorphic to $D_3\times \mathbb Z_2$?
Yes, $A_4$ has no element of order $6$, while $D_3\times Z_2$ does. You can use the same idea with the second question: what are the orders of the elements of $H$ and of $Z_4$?
Apr
16
comment Semidirect product.
Since $z$ has order $2$, you'll need the relation $z^2 = e$. But you also need relations describing the action of $z$ on the direct product of the dihedral groups: $z^{-1}x_1z = x_2, z^{-1}y_1z = y_2$.
Apr
13
comment Internal Direct Product
Well, you're almost there. You've shown that your subgroups $H$ and $K$ must be abelian, so what can you say about their direct product?
Apr
10
answered Are free products of finite cyclic groups perfect?