# Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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### There cannot be a group of units of order $14$

I proved the following claim: There is no $n$ such that $U(n)$ has order $14$. Please could someone tell me if my proof is correct? $U(n)$ is the group of units modulo $n$ and $\varphi$ is the ...
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### Short exact sequence of groups, is it possible to construct an associated fibration of spaces?

Let$$0 \to R \to F \to G \to 0$$be a short exact sequence of groups. Is it possible to construct an associated fibration of spaces$$K(R, 1) \to K(F, 1) \to K(G, 1)?$$
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### Convincing normal subgroup proof?

I wrote the following proof on an exam, I was wondering if it makes sense. Question: let $H$ be a subgroup of $G$, written in multiplicitive notation. Prove that if the coset multiplication defined ...
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### Let $G$ be a $2$-group and suppose the centralizer of some element of order two has order at most four, then $G$ has maximal class

Let $G$ be a $2$-group of order $|G| \ge 4$ and $H \le G$ be a subgroup of order $2$, i.e. $|H| = 2$. Suppose we have $|C_G(H)| \le 4$. Then $G$ has maximal class. Do you know a proof of this fact? I ...
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### Normalizers of subgroups of Sylow $p$-subgroups

I am wondering whether there is an easy example of a finite group $G$ with a Sylow $p$-subgroup $P$ and a subgroup $Q\leq P$ such that the normalizer $N_P(Q)$ of $Q$ in $P$ is NOT a Sylow $p$-subgroup ...
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### For which $p$ is $G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$ a subgroup of $S_{12}$?

Let $p$ be a prime number. I've to figure out for which $p$ $$G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$$ is a subgroup of $S_{12}$. I realize that we have to ...
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### How to use a proof by contradiction in group theory!

A group $G$ is said to be abelian-by-finite if it has a normal subgroup $H$ of finite index in $G$. I want to prove the following statement: "If a group $G$ has property $\mathcal P$ then it is ...
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### Restriction of automorphism to the generalized Fitting subgroup

for my master's thesis I am going through M.Hertweck and W.Kimmerle's article on Coleman automorphisms. I encountered the following reasoning, which I can't seem to follow. We know the following on ...
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### Two definitions of nilpotence

I don't understand why the following two definitions of nilpotence are equivalent: Definition 1. $G$ is $0$-step nilpotent if $G=\{e\}$. G is $k+1$th-step nilpotent if G is not $k$-step nilpotent, ...
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### How to prove $\operatorname{ord}_G(a^k) = r/\gcd(r,k)$?

If $G$ is a finite group and $a \in G$ an element with $\operatorname{ord}_G(a) = r$, then $\operatorname{ord}_G(a^k) = r/\gcd(r,k)$. I know that this statement is correct, but how can one prove it?
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### Is it possible for a group to be a finite union of subgroups of infinite index?

Just restating the title: Does there exist a group $G$ and subgroups $H_1, \ldots, H_k$ so that $[G:H_i]$ is infinite for each $i = 1, 2, \ldots, k$, and $G = H_1 \cup \cdots \cup H_k$? If $G$ is a ...
### Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime.
Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime where $Z(G)$ is the center of $G$ and $|G:Z(G)|$ is the index of $Z(G)$ in $G$. This was in a test that I had recently but I was not able ...