query: Cayley theorem to prove that a group of order $2^mk$, $k$ odd, can't be simple
positive_passages.docid: 25404
positive_passages.text: Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
negative_passages.docid: 134000
negative_passages.text: Proof by Strong Induction: $n = 2^a b,\, b\,$ odd, every natural a product of an odd and a power of 2
negative_passages.docid: 128834
negative_passages.text: Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
negative_passages.docid: 70688
negative_passages.text: How can I prove that the additive group of rationals is not isomorphic to a direct product of two nontrivial groups?
negative_passages.docid: 87766
negative_passages.text: Using Cayley's Theorem to find an isomorphic subgroup.
negative_passages.docid: 62277
negative_passages.text: Cayley tables for two non-isomorphic groups of order 4.
negative_passages.docid: 118022
negative_passages.text: Show that a group of order $180$ is not simple.
negative_passages.docid: 47478
negative_passages.text: Show that any cyclic group of even order has exactly one element of order $2$
negative_passages.docid: 147746
negative_passages.text: Characterisation of the squares of the symmetric group
negative_passages.docid: 35124
negative_passages.text: Can every non-simple group $G$ be written as a semidirect product?
negative_passages.docid: 99205
negative_passages.text: Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
negative_passages.docid: 62996
negative_passages.text: A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
negative_passages.docid: 90329
negative_passages.text: Prove there is no simple group of order $729$
negative_passages.docid: 29717
negative_passages.text: Not sure how to go about this group theory proof
negative_passages.docid: 105139
negative_passages.text: Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center
negative_passages.docid: 116737
negative_passages.text: An element of a group $G$ is not conjugate to its inverse if $\lvert G\rvert$ is odd
negative_passages.docid: 41688
negative_passages.text: An elementary problem in Group Theory: the unique noncyclic group of order 4