query: Question on a proof of: If $H$ is a normal subgroup of $G$ and $|H| = 2$, show that $H \subseteq Z(G)$.
positive_passages.docid: 25832
positive_passages.text: Let $K$ be a normal subgroup of order 2 in group $G$, show that $K$ lies in the centre of $G$
negative_passages.docid: 95534
negative_passages.text: If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
negative_passages.docid: 35800
negative_passages.text: If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
negative_passages.docid: 33387
negative_passages.text: Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$
negative_passages.docid: 115204
negative_passages.text: Show that SO(n) is a normal subgroup of O(n)
negative_passages.docid: 118337
negative_passages.text: Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$
negative_passages.docid: 102622
negative_passages.text: K is a subgroup of G. Show that K is not a normal subgroup
negative_passages.docid: 98739
negative_passages.text: If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
negative_passages.docid: 44942
negative_passages.text: A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
negative_passages.docid: 46355
negative_passages.text: If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
negative_passages.docid: 122581
negative_passages.text: Prove or disprove that $G_1/H_1 \cong G_2/H_2$
negative_passages.docid: 77361
negative_passages.text: How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?
negative_passages.docid: 106277
negative_passages.text: Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
negative_passages.docid: 50706
negative_passages.text: $|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
negative_passages.docid: 129753
negative_passages.text: Prove that a subgroup which contains half of all elements is a normal subgroup.
negative_passages.docid: 70373
negative_passages.text: Intersection of normal subgroups proof
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.