Question

Let $P \in{\rm Syl}_p(G)$ and suppose $P$ is metacyclic. Suppose $(|G|, p^2 - 1) = 1$. Show that $G$ has a normal $p$ complement.

score 38 · Answer 1 · 2021-12-13 16:28:49

38

Views

Let $P \in{\rm Syl}_p(G)$ and suppose $P$ is metacyclic. Suppose $(|G|, p^2 - 1) = 1$. Show that $G$ has a normal $p$ complement.

Published on 13 Dec 2021 - 16:28

score 27 · Answer 2 · 2026-03-26 21:12:39

27

Views

Generator/generators for cyclic group $\mathbb Z[i]/(11-8i)$, given the following isomorphism:

Published on 26 Mar 2026 - 21:12

score 62 · Answer 3 · 2026-04-04 13:11:02

62

Views

Describe all of the homomorphisms from $\Bbb Z_{24}$ to $\Bbb Z_{18}$.

Published on 04 Apr 2026 - 13:11

score 81 · Answer 4 · 2021-12-13 22:58:28

81

Views

How to show that the group $S_3 \times C_2$ contains three element of order 3?

Published on 13 Dec 2021 - 22:58

score 110 · Answer 5 · 2026-04-04 14:50:55

110

Views

Showing that for $G = \langle x \rangle$ if $|G|$ is odd, then $\phi$ is an injection.

Published on 04 Apr 2026 - 14:50

score 154 · Answer 6 · 2026-03-30 20:43:01

154

Views

Proving Schur-Zassenhaus Theorem, with added assumption that $G/H$ is cyclic

Published on 30 Mar 2026 - 20:43

score 44 · Answer 7 · 2026-04-04 14:51:06

44

Views

How do I prove that the image of a cyclic group has the following property?

Published on 04 Apr 2026 - 14:51

score 75 · Answer 8 · 2021-12-25 19:30:04

75

Views

What are the values of $a$ and $b$ so that $\Bbb Z_2\times\Bbb Z_3\times\Bbb Z_4\times\Bbb Z_9$ is isomorphic with $\Bbb Z_a\times\Bbb Z_b$?

Published on 25 Dec 2021 - 19:30

score 191 · Answer 9 · 2021-12-27 07:22:08

191

Views

Frattini subgroup of a cyclic group which is not a p group for some prime p

Published on 27 Dec 2021 - 7:22

score 112 · Answer 10 · 2021-12-28 13:21:15

112

Views

cyclic representation of $L^∞(X, Ω, µ)$

Published on 28 Dec 2021 - 13:21

score 47 · Answer 11 · 2026-04-06 00:15:39

47

Views

Is this statement about an element in $\mathbb{Z}/n\mathbb{Z}$ correct?

Published on 06 Apr 2026 - 0:15

score 71 · Answer 12 · 2021-12-31 14:01:22

71

Views

Proof that all subgroups of $(\Bbb Z_{15},+)$ are cyclic and list all of its distinct groups

Published on 31 Dec 2021 - 14:01

score 216 · Answer 13 · 2026-03-28 11:34:24

216

Views

How many cyclic subgroups are in $\Bbb Z_8\oplus\Bbb Z_4\oplus\Bbb Z_2$?

Published on 28 Mar 2026 - 11:34

score 285 · Answer 14 · 2026-03-30 07:40:39

285

Views

Is the cyclic group $\langle x\rangle$ always a subgroup of $G$ for any $x\in G$?

Published on 30 Mar 2026 - 7:40

score 42 · Answer 15 · 2022-01-09 10:26:47

42

Views

How can this subgroup $H$ of a cyclic group $G=\langle x \rangle$ contain the identity element if $H= \{1x, 2x, 3x, .... \} $?

Published on 09 Jan 2022 - 10:26

List Question

Let $P \in{\rm Syl}_p(G)$ and suppose $P$ is metacyclic. Suppose $(|G|, p^2 - 1) = 1$. Show that $G$ has a normal $p$ complement.

Generator/generators for cyclic group $\mathbb Z[i]/(11-8i)$, given the following isomorphism:

Describe all of the homomorphisms from $\Bbb Z_{24}$ to $\Bbb Z_{18}$.

How to show that the group $S_3 \times C_2$ contains three element of order 3?

Showing that for $G = \langle x \rangle$ if $|G|$ is odd, then $\phi$ is an injection.

Proving Schur-Zassenhaus Theorem, with added assumption that $G/H$ is cyclic

How do I prove that the image of a cyclic group has the following property?

What are the values of $a$ and $b$ so that $\Bbb Z_2\times\Bbb Z_3\times\Bbb Z_4\times\Bbb Z_9$ is isomorphic with $\Bbb Z_a\times\Bbb Z_b$?

Frattini subgroup of a cyclic group which is not a p group for some prime p

cyclic representation of $L^∞(X, Ω, µ)$

Is this statement about an element in $\mathbb{Z}/n\mathbb{Z}$ correct?

Proof that all subgroups of $(\Bbb Z_{15},+)$ are cyclic and list all of its distinct groups

How many cyclic subgroups are in $\Bbb Z_8\oplus\Bbb Z_4\oplus\Bbb Z_2$?

Is the cyclic group $\langle x\rangle$ always a subgroup of $G$ for any $x\in G$?

How can this subgroup $H$ of a cyclic group $G=\langle x \rangle$ contain the identity element if $H= \{1x, 2x, 3x, .... \} $?

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