Question

Count the subgroups of order $p^2$ in $\mathbb{Z}_{p^3}\oplus\mathbb{Z}_{p^2}$

score 34 · Answer 1 · 2021-12-13 14:48:11

34

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Count the subgroups of order $p^2$ in $\mathbb{Z}_{p^3}\oplus\mathbb{Z}_{p^2}$

Published on 13 Dec 2021 - 14:48

score 98 · Answer 2 · 2021-12-20 20:39:58

98

Views

Under what conditions can we say that $G/N \cong S \implies G \cong N \times S$?

Published on 20 Dec 2021 - 20:39

score 82 · Answer 3 · 2022-01-06 12:33:22

82

Views

For $P$ the set of all primes, ${\rm Hom}(\Bbb Z,\sum_{p\in P}\Bbb Z_p)\not\cong\prod_{p\in P}{\rm Hom}(\Bbb Z,\Bbb Z_p)$

Published on 06 Jan 2022 - 12:33

score 212 · Answer 4 · 2022-01-09 17:00:51

212

Views

Is the given condition necessary for $HK$ to be isomorphic to $H\times K$?

Published on 09 Jan 2022 - 17:00

score 59 · Answer 5 · 2022-01-09 18:14:55

59

Views

Define a binary operation on {e,(12)}×{e,(123),(132)} so that it becomes isomorphic to $S_3$

Published on 09 Jan 2022 - 18:14

score 96 · Answer 6 · 2022-01-09 19:08:32

96

Views

If $HK$ is a subgroup but not equal to $G$ (where $H,K<G$), then are $H$ and $K$ normal in $HK$?

Published on 09 Jan 2022 - 19:08

score 187 · Answer 7 · 2026-03-28 06:40:23

187

Views

Two exercises by Robinson on supersolvable groups seem to contradict.

Published on 28 Mar 2026 - 6:40

score 46 · Answer 8 · 2022-02-03 14:55:57

46

Views

A direct product $G=\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4$ of the cyclic group $\mathbb{Z}_4$ of order $4$.

Published on 03 Feb 2022 - 14:55

score 121 · Answer 9 · 2022-02-24 10:15:15

121

Views

Does there exist some isomorphism from $Q_8$ to $\Bbb Z_4 \times\Bbb Z_2$?

Published on 24 Feb 2022 - 10:15

score 54 · Answer 10 · 2022-02-25 02:27:23

54

Views

Show subgroup isomorphic to direct product

Published on 25 Feb 2022 - 2:27

score 115 · Answer 11 · 2022-02-25 17:28:06

115

Views

Two competing definitions of the direct sum of vector spaces

Published on 25 Feb 2022 - 17:28

score 48 · Answer 12 · 2022-02-26 08:18:12

Show that there is no subgroup B of $\mathbb Z ^2$ such that $\mathbb Z ^2 / B \cong \mathbb Z_3 \times \mathbb Z_6 \times \mathbb Z_{21}$

score 177 · Answer 13 · 2026-03-26 06:12:11

177

Views

Trying to understand the differences between $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$

Published on 26 Mar 2026 - 6:12

score 117 · Answer 14 · 2022-03-08 03:15:29

117

Views

Cohomology ring of disjoint union

Published on 08 Mar 2022 - 3:15

score 66 · Answer 15 · 2022-03-14 09:36:37

66

Views

Gradings and direct sums notation

Published on 14 Mar 2022 - 9:36

List Question

Count the subgroups of order $p^2$ in $\mathbb{Z}_{p^3}\oplus\mathbb{Z}_{p^2}$

Under what conditions can we say that $G/N \cong S \implies G \cong N \times S$?

For $P$ the set of all primes, ${\rm Hom}(\Bbb Z,\sum_{p\in P}\Bbb Z_p)\not\cong\prod_{p\in P}{\rm Hom}(\Bbb Z,\Bbb Z_p)$

Is the given condition necessary for $HK$ to be isomorphic to $H\times K$?

Define a binary operation on {e,(12)}×{e,(123),(132)} so that it becomes isomorphic to $S_3$

If $HK$ is a subgroup but not equal to $G$ (where $H,K<G$), then are $H$ and $K$ normal in $HK$?

Two exercises by Robinson on supersolvable groups seem to contradict.

A direct product $G=\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4\times\mathbb{Z}_4$ of the cyclic group $\mathbb{Z}_4$ of order $4$.

Does there exist some isomorphism from $Q_8$ to $\Bbb Z_4 \times\Bbb Z_2$?

Show subgroup isomorphic to direct product

Two competing definitions of the direct sum of vector spaces

Show that there is no subgroup B of $\mathbb Z ^2$ such that $\mathbb Z ^2 / B \cong \mathbb Z_3 \times \mathbb Z_6 \times \mathbb Z_{21}$

Trying to understand the differences between $\mathbb{Z}_2 * \mathbb{Z}_2$ vs $\mathbb{Z}_2 \times \mathbb{Z}_2$

Cohomology ring of disjoint union

Gradings and direct sums notation

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