# Tagged Questions

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### Is the map an automorphism?

Please verify the following proof or comment on how would you have proven it. Suppose $q = p^2$ and we have $f: \mathbb{F}_{q} \to \mathbb{F}_{q}$ where $f(a) = a.a^p$ Let $a.a^p = b.b^p$ ...
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### regarding the groups that how they form and what may be the practicaliy operations those taken to define the groups [on hold]

I studied the group theory and also the ring theory that some algebraic structures those satisfying some algebraic operations are known so. But some cases I found that some structures form a group ...
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### $N$ a normal subgroup of $G$ with $|G|$ odd and $|N|=5$ satisfies $N \subset Z(G)$

Exercise Let $N$ be a normal subgroup of $G$. Suppose that $|N|=5$ and that $|G|$ is odd. Show that $N \subset Z(G)$. I am sorry for not writing any work of mine but I really don't know where to ...
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### Generalization of Lattice Theorem

I've shown that Lattice Theorem is still true if I substitute the projection to the quotient with a generic surjective homomorphism, i.e. if $G$ and $G'$ are two groups and $f:G\to G'$ is a surjective ...
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### $x \rightarrow x^n$ is a group automorphism of a finite abelian group G [on hold]

How do we prove that the map $\phi:G \rightarrow G$ defined by $\phi(x) = x^n$ for some $n \geq 0$ is a group automorphism of $G$ if $\gcd(|G|,n)=1$?
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### How to prove that a subgroup of a group is normal based on generating sets?

I apologize if this is a duplicate question, but I read online that one method by which to show that a subgroup is normal is by means of generating sets (if both groups have known presentations). In ...
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### Order of a permutation

What does the order of a permutation actually mean? I accept the fact that it is the l.c.m. of the lengths of the cycles in its cycle decomposition, but I don't really have an intuition for what the ...
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### Realize groups as unit group of a ring

Let $A$ be a ring, $G$ be a group, and $f:A^{\times} \rightarrow G$ be a group homomorphism. Is there any ring $B$ and ring homomorphism $\varphi:A \rightarrow B$ such that $G$ is subgroup of ...
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### Constructing group homomorphisms.

Let $G$ and $H$ be groups, and suppose I want to construct a group homomorphism $\phi$ between them. From what I know, I just need to send each element $x \in G$ to an element $y \in H$ such that the ...
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### Property of group G with $|G|=2n$ with $n$ elements of order $2$ (Sylow theorem application)

Suppose $G$ is a group such that $|G|=2n$, $G$ has $n$ elements of order $2$ and the rest of the elements form a subgroup $H$. Show that $H \lhd G$ and $n$ is odd. I am pretty lost with this ...
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### How do I get the generators of a group formed by combining two groups with known generators?

Consider two groups, $G$ and $H$, with generating sets $S$ and $T$, respectively. (That is, $G=\langle S \rangle$ and $H=\langle T \rangle$.) Let us say that we can represent elements of both $G$ ...
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### Self normalising p sylow

When are p-sylow subgroups self normalising? I know, for example, that if the group has order $p^2q^2$ then the p-sylow subgroups are self-normalising if there are $q^2$ of them. I just don't know ...
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### Determining if a set is a group

Let $S=\lbrace x+y\sqrt2 : x,y\in \mathbb R \rbrace$ \ $\lbrace0\rbrace$. Justify whether $S$, together with traditional multiplication, is a group. I've verified that the set is closed under the ...
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### How many automorphisms does $S_3\times S_3$ have?

I've shown that $|\text{Aut}(S_3\times S_3)|\ge 72$, how can I show that $|\text{Aut}(S_3\times S_3)|\le 72$ ?
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### Let $H\le G$ s. t. whenever $Ha≠Hb$ then $aH≠bH$. Prove that $gHg^{−1}\le H\;$ $\forall g\in G$. [on hold]

Suppose that H is a subgroup of G such that whenever$Ha \ne Hb$ then $aH \ne bH$. Prove that $gHg^{-1} \subseteq H$ for all g in G.
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### Number of Sylow 2-subgroups of a special linear group

Find the number of Sylow $2$-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo $3$). I think it will be $1$. But I failed to prove it using the counting principle. It has $4$ ...
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### Orbit and stabaliser of $2\times2 * 2\times1$ matrices

I have the group action of matrix multiplication, meaning: $g((x,y))=\begin{pmatrix}a&0\\0&b\end{pmatrix}$$\begin{pmatrix}x\\ y\end{pmatrix}$ ...
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### If $X$ generates $\Bbb{Q}$ then $X\setminus\{x\}$ also generates $\Bbb{Q}$ [duplicate]

If $X$ is a generator subset of $\Bbb{Q}$ then for $x\in X$, $X\setminus\{x\}$ also generates $\Bbb{Q}$. Clearly if I can express $x$ as a combination of the remaining generators we are done. ...
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### Normal subgroup of a finite group

Let $G=\langle X_1,X_2\rangle$. Can we say that if $X_1$ or $X_2$ is a normal subgroup of $G$, then $G=X_1X_2$?
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### what are the p-orbits in the decomposition of Σ into p-orbits.

At the bottom in the proof of slow 2, what's the meaning of "restrict the action of G on Σ to an...on Σ"? Since I can't understand that so I don't know what are the p-orbits in the decomposition of Σ ...
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### Is orbit a group?

Let a group $G$ act on a set $S$, and let $s$ be an element in $S$. The identity of the orbit of $s$ is $s$ itself and if $a$, $b$ are in the orbit of $s$, then $a b = g_1 g_2 s$, where $g_1$, $g_2$ ...
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### Rapid question on minimal normal subgroups

Why does $N$ finite group, $M\unlhd N$ imply that $M$ contains a minimal normal subgroup of $N$? If $M$ is itself minimal in $N$ we have finished. But if not, we know that there exists a minimal ...
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### Product of two stabilizers of transitive group action is proper subset of G?

Suppose $G$ is a finite group and G acts transitively on some set $X$. Let $a$ and $b$ be two distinct elements of $X$ and $G_{a}$ and $G_{b}$ be stabilizers of $a$ and $b$ respectively.Show that ...
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### What is the group $_nG$?

I've seen the notation $_nG$, where $G$ is a group and $n > 1$ an integer. What does this notation mean? I suspect it has to do with torsion somehow. Maybe a notation for the $n$-torsion subgroup ...
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### solvable group and derived series [duplicate]

Can someone prove that group is solvable if and only if its derived series terminate on a trivial group? http://en.wikipedia.org/wiki/Solvable_group
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### Proving that a set is a group under addition

To show that a set $G$ is a group under addition, do we first need to show that $G$ is closed under addition, or is that implied by proving the three properties of a group, namely there exists an ...
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### Can you explain more specifically about there is no simple group of size 96

For the solution,I know that the number of Sylow-$2$ subgroups, $n_2=3$, and we can find a subgroup of order $32$,then $G$ can act on the left cosets of $H=P_2$, then it gives a map from $G$ to $S_3$, ...
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### Quotient group $G/\{1\}=G$ if $1$ is the identity element of $G$

Is it true that quotient group $G/\{1\}=G$? Or isomorphic to $G$?
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### Why are centers, centralizers and normalizers called that way?

I know what they are, but where do the names come from?
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### Upper bound for the number of generators of a group

Let $H\leq G$ and $x\in G$. If $H$ is generated by at most $n$ elements, prove that $\langle H,x\rangle$ is generated by at most $n+1$ elements. This is intuitively obvious but when I try to ...
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### Direct product of two groups.

Let $N$ be a minimal normal subgroup of $G$. Also let $N$ and $\frac{G}{N}$ are non-abelian simple. Can we say that $G=N\times A$ where $A$ is a non-abelian simple subgroup of $G$?
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### non-abelian simple group

Let G be a non-solvable group and $(\frac{G}{Z(G)})^{'}=\frac{G}{Z(G)}$. Can we say that there is a normal subgroup $N$ of $G$ with property $Z(G)\leq N$ such that $\frac{G}{N}$ is a non-abelian ...
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### Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?

Is the following statement true or false: If $G$ is a group with the property that $g=g^{-1}$ for all $g \in G$, then $G$ is abelian. I believe it is false since I know that abelian ...
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### If $N$ normal subgroup of $G$ and $M$ normal subgroup of $G$ prove $MN$ is a subgroup [closed]

If $N\vartriangleleft G$ and $M\vartriangleleft G$ and $MN = \{mn | m \in M, n \in N\}$, prove that $MN$ is a subgroup of $G$ and that $MN\vartriangleleft G$
I have been working on this problem for an hour now and gotten nowhere: Let $G$ be any group and $A(G)$ the set of all 1-1 mappings of $G$, as a set, onto itself. Define $L_a : G \rightarrow G$ by ...
### If $G$ is a finite group whose $p$-Sylow subgroup $P$ lies in its center, then there is a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$
If $G$ is a finite group and its $p$-sylow subgroup $P$ lies in the center of $G$, prove that there exists a normal subgroup $N$ of $G$ with $P\cap N=\{e\}$ and $PN=G$