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 1d answered Is there a number $n$, such that there are $22$ groups of order $n$? Nov 24 comment What is an equivalent version of maple software in linux? There is a free CAS called Sage that works on Linux. It is quite different from Maple, so I wouldn't call it equivalent. You can't run your Maple code in it, as far as I am aware. But it is quite comprehensive in its own right, and seems to be popular. Nov 24 comment What is an equivalent version of maple software in linux? Maple is available on Linux. Nov 20 comment What is the smallest number $d$, not a power of $2$, such that the number of groups with order d is unknown? I don't know whether the number of groups of order $2250 = 2\cdot 3^2\cdot 5^3$ is known, but that is the smallest number $d$ (not a power of $2$) for which Maple does not currently know the number of groups of order $d$. Nov 11 answered Understanding how a subgroup is normal if and only if the relation $\equiv$ (N) respects products and inverses Nov 11 answered If $H\leqslant G$ and $K\leqslant G$, is it always true that $HK\leqslant G$? Nov 9 answered Generating random groups satisfying certain conditions Nov 4 comment How to represent Nielsen automorphisms as permutations? Arbitrary automorphisms of $F_n$ are not always expressible as compositions of permutation automorphisms. In particular, a Nielsen automorphism that inverts a generator is not expressible in this way. Put another way, the permutation automorphisms generate a proper subgroup of $\operatorname{Aut} F_n$. Oct 30 comment Let $G=S_4$. Show that $G$ is isomorphic to $J(G)$ The notation is not standard, but I would guess that $J(G)$ is the inner automorphism group of $G$. That would, at least, render the statement true. Oct 27 comment Example of a group that centralizers of every non-identity elements is not abelian A quick computer search with Maple shows that there is an example of order $32$. It is SmallGroup(32,49). Oct 26 comment $G$ is a group and $Z(G)$ its center. $f\colon G\to G$ is an automorphism of $G$. Show that if $x$ is in $Z(G)$, then $f(x)$ is also in $Z(G)$. @likelikelike From $xy = yx$, apply $f$ to get $f(xy) = f(yx)$ and then, as $f$ is a homomorphism, it follows that $f(x)f(y) = f(y)f(x)$. Now, $f(y)$ is essentially arbitrary in $G$ because $f$ is surjective. Oct 26 answered $G$ is a group and $Z(G)$ its center. $f\colon G\to G$ is an automorphism of $G$. Show that if $x$ is in $Z(G)$, then $f(x)$ is also in $Z(G)$. Oct 26 answered Proof about cyclic group with prime order and the relation between the cardinality of the set on which it acts and the fixed points of the set Oct 9 answered Question about 3rd Sylow theorem Oct 5 comment Is it possible to construct a field larger than the complex numbers? Sure, construct the field $\mathbb{C}(z)$ of rational functions in $z$ with complex coefficients. Oct 5 comment If every sylow subgroup of $H$ is cyclic, is $H$ soluble group ? Yes, in fact $H$ is metacyclic. The finite groups with cyclic Sylow subgroups are precisely the finite metacyclic groups. A proof can be found in Robinson, among other sources. How $H$ is situated in $G$ appears not to be relevant to the question, unless I've misunderstood something. Oct 3 answered Maple zeroth entry of list or some other data structure? Sep 21 revised Palindrome Induction Proof Fixed spelling 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