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1d
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.
1d
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.
1d
comment Trying to show $|ab|$ divides lcm$(|a|,|b|)$
Try looking at the set of distinct (cyclic) subgroups of order $p$.
1d
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?
Apr
10
comment Are free products of finite cyclic groups perfect?
It's not perfect. There is an epimorphism onto the cyclic group of order $6$, which is $\mathbb{Z}_2\oplus\mathbb{Z}_3$.
Apr
10
comment Find all sylow subgroups of$ D_{2p^k}$, where p is prime
If $p=2$, then $D_{2p^k}$ is a $2$-group, meaning that it has only a Sylow $2$-subgroup equal to itself.
Apr
10
answered If $f(g) = g^k$ is a homomorphism on a finite group $G$ and $k < |G|$ does not divide $|G|$, must $G$ be abelian?
Apr
8
revised property of quandle
added 26 characters in body
Apr
8
answered property of quandle
Apr
7
comment Permutations $S_n$
This previous question might be of some help to you.
Apr
6
comment Find if relation is reflexive, symmetric or transitive
It looks like another "James" has essentially summarised what you need to do.
Apr
6
comment Find if relation is reflexive, symmetric or transitive
It's reflexive and symmetric, but not transitive.
Apr
6
comment Show that c belongs to the commutator subgroup
Too long for a comment; please see additional edits.
Apr
6
revised Show that c belongs to the commutator subgroup
added 607 characters in body
Apr
6
comment Show that c belongs to the commutator subgroup
Yes that is the correct map. Just evaluate the image of $c$ in $\overline{G} = G/G^{\prime}$, noting that this is an abelian group.