Understanding direct and semi direct products through notations

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Can someone please help to understand and identify the following groups?




  1. $G_1 = langle a, b mid a^p^2 = b^q = 1, b a b^-1 = a^i, operatornameord_p^2(i) = q rangle$.

  2. A family of groups given by,
    $$
    langle
    a, b, c
    mid
    a^p = b^p = c^q = 1,
    c a c^-1 = a^i,
    c b c^-1 = b^i^t,
    ab = ba,
    operatornameord_p(i) = q
    rangle
    $$
    where $p, q$—distinct primes, $p > q$ and $q mid p - 1$.

(Original image here.)




Can the first group be written as,
$$
mathbbZ_p^2 rtimes_varphi mathbbZ_q
$$
and the second group as,
$$
(mathbbZ_p times mathbbZ_p) rtimes mathbbZ_q.
$$
What does $operatornameord_p^2(i) = q$, $operatornameord_p(i) = q$ mean?




group-theory finite-groups semidirect-product




share|cite|improve this question






















  • $textord_p(i)=q$ means that the multiplicative order of $i$ modulo $p$ is $q$
    – rbird
    Aug 26 at 10:05











  • The answers to your first two questions are both yes.
    – Derek Holt
    Aug 26 at 10:09










  • Thanks. Can I regard them as an extension of the semidirect product between $Z_p$ and $Z_q$?
    – Buddhini Angelika
    Aug 26 at 10:35










  • Can I say that the above two semidirect products are the direct products?
    – Buddhini Angelika
    Aug 26 at 17:14










  • No they are not direct products. If they were direct products then they would be abelian groups, which they are not.
    – Derek Holt
    Aug 26 at 18:59














up vote
1
down vote

favorite












Can someone please help to understand and identify the following groups?




  1. $G_1 = langle a, b mid a^p^2 = b^q = 1, b a b^-1 = a^i, operatornameord_p^2(i) = q rangle$.

  2. A family of groups given by,
    $$
    langle
    a, b, c
    mid
    a^p = b^p = c^q = 1,
    c a c^-1 = a^i,
    c b c^-1 = b^i^t,
    ab = ba,
    operatornameord_p(i) = q
    rangle
    $$
    where $p, q$—distinct primes, $p > q$ and $q mid p - 1$.

(Original image here.)




Can the first group be written as,
$$
mathbbZ_p^2 rtimes_varphi mathbbZ_q
$$
and the second group as,
$$
(mathbbZ_p times mathbbZ_p) rtimes mathbbZ_q.
$$
What does $operatornameord_p^2(i) = q$, $operatornameord_p(i) = q$ mean?




group-theory finite-groups semidirect-product




share|cite|improve this question






















  • $textord_p(i)=q$ means that the multiplicative order of $i$ modulo $p$ is $q$
    – rbird
    Aug 26 at 10:05











  • The answers to your first two questions are both yes.
    – Derek Holt
    Aug 26 at 10:09










  • Thanks. Can I regard them as an extension of the semidirect product between $Z_p$ and $Z_q$?
    – Buddhini Angelika
    Aug 26 at 10:35










  • Can I say that the above two semidirect products are the direct products?
    – Buddhini Angelika
    Aug 26 at 17:14










  • No they are not direct products. If they were direct products then they would be abelian groups, which they are not.
    – Derek Holt
    Aug 26 at 18:59












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Can someone please help to understand and identify the following groups?




  1. $G_1 = langle a, b mid a^p^2 = b^q = 1, b a b^-1 = a^i, operatornameord_p^2(i) = q rangle$.

  2. A family of groups given by,
    $$
    langle
    a, b, c
    mid
    a^p = b^p = c^q = 1,
    c a c^-1 = a^i,
    c b c^-1 = b^i^t,
    ab = ba,
    operatornameord_p(i) = q
    rangle
    $$
    where $p, q$—distinct primes, $p > q$ and $q mid p - 1$.

(Original image here.)




Can the first group be written as,
$$
mathbbZ_p^2 rtimes_varphi mathbbZ_q
$$
and the second group as,
$$
(mathbbZ_p times mathbbZ_p) rtimes mathbbZ_q.
$$
What does $operatornameord_p^2(i) = q$, $operatornameord_p(i) = q$ mean?




group-theory finite-groups semidirect-product




share|cite|improve this question














Can someone please help to understand and identify the following groups?




  1. $G_1 = langle a, b mid a^p^2 = b^q = 1, b a b^-1 = a^i, operatornameord_p^2(i) = q rangle$.

  2. A family of groups given by,
    $$
    langle
    a, b, c
    mid
    a^p = b^p = c^q = 1,
    c a c^-1 = a^i,
    c b c^-1 = b^i^t,
    ab = ba,
    operatornameord_p(i) = q
    rangle
    $$
    where $p, q$—distinct primes, $p > q$ and $q mid p - 1$.

(Original image here.)




Can the first group be written as,
$$
mathbbZ_p^2 rtimes_varphi mathbbZ_q
$$
and the second group as,
$$
(mathbbZ_p times mathbbZ_p) rtimes mathbbZ_q.
$$
What does $operatornameord_p^2(i) = q$, $operatornameord_p(i) = q$ mean?





group-theory finite-groups semidirect-product





share|cite|improve this question













share|cite|improve this question




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edited Aug 26 at 10:01









Jendrik Stelzner

7,58221037




7,58221037










asked Aug 26 at 9:50









Buddhini Angelika

485




485











  • $textord_p(i)=q$ means that the multiplicative order of $i$ modulo $p$ is $q$
    – rbird
    Aug 26 at 10:05











  • The answers to your first two questions are both yes.
    – Derek Holt
    Aug 26 at 10:09










  • Thanks. Can I regard them as an extension of the semidirect product between $Z_p$ and $Z_q$?
    – Buddhini Angelika
    Aug 26 at 10:35










  • Can I say that the above two semidirect products are the direct products?
    – Buddhini Angelika
    Aug 26 at 17:14










  • No they are not direct products. If they were direct products then they would be abelian groups, which they are not.
    – Derek Holt
    Aug 26 at 18:59
















  • $textord_p(i)=q$ means that the multiplicative order of $i$ modulo $p$ is $q$
    – rbird
    Aug 26 at 10:05











  • The answers to your first two questions are both yes.
    – Derek Holt
    Aug 26 at 10:09










  • Thanks. Can I regard them as an extension of the semidirect product between $Z_p$ and $Z_q$?
    – Buddhini Angelika
    Aug 26 at 10:35










  • Can I say that the above two semidirect products are the direct products?
    – Buddhini Angelika
    Aug 26 at 17:14










  • No they are not direct products. If they were direct products then they would be abelian groups, which they are not.
    – Derek Holt
    Aug 26 at 18:59















$textord_p(i)=q$ means that the multiplicative order of $i$ modulo $p$ is $q$
– rbird
Aug 26 at 10:05





$textord_p(i)=q$ means that the multiplicative order of $i$ modulo $p$ is $q$
– rbird
Aug 26 at 10:05













The answers to your first two questions are both yes.
– Derek Holt
Aug 26 at 10:09




The answers to your first two questions are both yes.
– Derek Holt
Aug 26 at 10:09












Thanks. Can I regard them as an extension of the semidirect product between $Z_p$ and $Z_q$?
– Buddhini Angelika
Aug 26 at 10:35




Thanks. Can I regard them as an extension of the semidirect product between $Z_p$ and $Z_q$?
– Buddhini Angelika
Aug 26 at 10:35












Can I say that the above two semidirect products are the direct products?
– Buddhini Angelika
Aug 26 at 17:14




Can I say that the above two semidirect products are the direct products?
– Buddhini Angelika
Aug 26 at 17:14












No they are not direct products. If they were direct products then they would be abelian groups, which they are not.
– Derek Holt
Aug 26 at 18:59




No they are not direct products. If they were direct products then they would be abelian groups, which they are not.
– Derek Holt
Aug 26 at 18:59















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