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seen Jul 23 at 4:43

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Jul
13
accepted A counter-example to Dini's Theorem (after removing a hypothesis)
Jul
10
comment Does every group act faithfully on some group?
No need to be sorry at all :) I do like your free group example! +1
Jul
10
answered Does every group act faithfully on some group?
Jul
10
comment Does every group act faithfully on some group?
After seeing your edit, I realize that the misunderstanding lies in the usage of the term "acting faithfully". I think asking for injective group homomorphism $G\to\operatorname{Aut}(H)$ is a more unambiguous question. In any case, thanks for taking your time to answer the question :)
Jul
10
comment Does every group act faithfully on some group?
Let $G$ be a group. Then we have left-multiplication action: for each $g\in G$, we have a map $\sigma_{g}: G\to G$ defined by $\sigma_{g}(x)=gx$. It is action of $G$ on the set $G$. Note that $\sigma_{g}$ is a set-theoretic bijection, but is not a group isomorphism. So the left-multiplication defines a map $\sigma: G\to S_{|G|}$ and not $\sigma: G\to\operatorname{Aut}(G)$.
Jul
10
comment Does every group act faithfully on some group?
"Every group acts faithfully on itself by left multiplication." This is action of a group on itself as a set! Left multiplication map is not a group homomorphism.
Jul
10
comment A counter-example to Dini's Theorem (after removing a hypothesis)
@DavidMitra: Thanks David! Your answer in that thread is very helpful.
Jul
10
answered A counter-example to Dini's Theorem (after removing a hypothesis)
Jul
10
asked A counter-example to Dini's Theorem (after removing a hypothesis)
Jul
10
accepted Does every group act faithfully on some group?
Jul
9
asked Does every group act faithfully on some group?
Jul
2
awarded  Curious
Jul
2
awarded  Inquisitive
Jul
2
accepted Decomposing an element into product of elements of finite order
Jul
1
comment Decomposing an element into product of elements of finite order
Interesting! So the abelian case has the complete answer. Thanks for the elaboration.
Jul
1
comment Decomposing an element into product of elements of finite order
Sweet! Just to fill in the obvious line: if such a decomposition exists, then $|\det(x)|=|\det(gh)|=|\det(g)\det(h)| = |\det(g)||\det(h)| = 1$. Thanks so much, Hagen!
Jul
1
revised Decomposing an element into product of elements of finite order
edited body
Jul
1
comment Decomposing an element into product of elements of finite order
@user1729: Thanks! That's at least something :)
Jul
1
asked Decomposing an element into product of elements of finite order
Jun
5
awarded  Nice Question