Specific automorphisms on a 2-generated p-group of class two Nor Haniza Sarmin and Yasamin Barakat Citation: AIP Conference Proceedings 1557, 41 (2013); doi: 10.1063/1.4823871 View online: http://dx.doi.org/10.1063/1.4823871 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1557?ver=pdfcov Published by the AIP Publishing
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Specific Automorphisms On A -Generated -Group Of Class Two Nor Haniza Sarmin1 and Yasamin Barakat2 1,2
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Malaysia. 2
Islamic Azad University-Ahvaz Branch, Ahvaz, Iran.
Abstract. Let ܩbe a finite two generated -group of nilpotency class two, where is a prime number. If ߶ሺܩሻ is the Frattini subgroup of ܩ, then ܩΤ߶ሺܩሻ is a vector space of dimension two over the field ԺΤԺ. In this paper, we determine those automorphisms on ܩwhich induce identity transformation on ܩΤ߶ሺܩሻ. We use the most updated classification given for ܩto find these automorphisms. The results of this study are applicable in classifying ݐݑܣሺܩሻ, the automorphism group of ܩ. Keywords: Automorphism; two generated; -group; nilpotency of class . PACS: 02.20.-a
INTRODUCTION
GROUPS OF CLASS TWO
Let be a prime and ܩbe a finite non-abelian group of nilpotency class two. Suppose ܩis generated by two elements ܽ and ܾ. Then ܩƍ, the derived subgroup of ܩ, is cyclic and generated by ሾܽǡ ܾሿ ൌ ܽିଵ ܾ ିଵ ܾܽ [1]. If ȁܩƍȁ ൌ ఊ for some positive integer ߛ, then the center of ܩis of the form ܼሺܩሻ ൌ ം ം ܽۃ ǡ ܾ ǡ ሾܽǡ ܾሿ[ ۄ2]. It is known that Frattini subgroup of ܩ, ߶ሺܩሻ, is the intersection of all maximal subgroups of ܩ. Let ܯ be an arbitrary maximal subgroup of ܩ. Since ܩis a group we have ȁܩǣ ܯȁ ൌ . Hence, ݃ ߶ אሺܩሻ for all ݃ ܩ אand also ܩƍ ߶ كሺܩሻ. This implies that ߶ሺܩሻ ൌ ܽۃ ǡ ܾ ǡ ሾܽǡ ܾሿ[ ۄ3]. Furthermore, we can conclude ܩҧ ൌ ܩΤ߶ሺܩሻ ൌ ߶ܽۃሺܩሻǡ ܾ߶ሺܩሻ ۄis a vector space of dimension two over ԺΤԺ, the field of integers modulo . Moreover, every automorphism ߠ on ܩ induces an automorphism ߠҧ on ܩҧ defined by ߠҧሺܽ߶ሺܩሻሻ ൌ ߠሺܽሻ߶ሺܩሻ and ߠҧ ሺܾ߶ሺܩሻሻ ൌ ߠሺܾሻ߶ሺܩሻ. In this study, we specify those ߠ's that induce the identity transformation on ܩҧ . As a matter of fact, we use the most updated classification of ܩto classify ܣథሺீሻ ൌ ሼߠ ݐݑܣ אሺܩሻ ߠ ҧ ൌ ݅ሽ. The results of this study can be used to determine ݐݑܣሺܩሻ, since ݐݑܣሺܩሻΤܣథሺீሻ is isomorphic to a subgroup of ܮܩሺʹǡ ሻ, the group of all non-degenerated transformations on ܩҧ .
Let ܪǡ ܭbe two arbitrary subgroups of a group ܩ. Then, their commutator, given as ሾܪǡ ܭሿ ൌ ۃሼ݄ିଵ ݇ ିଵ ݄݇ ܪ݄߳ ǡ ݇߳ܭሽ ۄis a subgroup of ܩ. Since ݄ିଵ ݇ ିଵ ݄݇ ൌ ͳ implies ݄݇ ൌ ݄݇, hence, ܪand ܭcommutes if and only if ሾܪǡ ܭሿ ൌ ሼͳሽ. Let ଵ ሺܩሻ ൌ ܩand ାଵ ሺܩሻ ൌ ሾܩǡ ሺܩሻሿ for ݅ ͳ. A group ܩis called nilpotent of class ݊ if ାଵ ሺܩሻ ൌ ሼͳሽ for some positive integer ݊. Therefore, ݊ ൌ ͳ leads to ܩƍ ൌ ሾܩǡ ܩሿ ൌ ሼͳሽ or ܩis abelian. In case ݊ ൌ ʹ, ሾܩǡ ܩƍ ሿ ൌ ሼͳሽ implies that ܩƍ ܼ كሺܩሻ. Consequently, for any ݔǡ ݕǡ ݖin a group ܩof nilpotency class ʹ and ݉ אԺ, the following equations hold: x ሾݔǡ ݖݕሿ ൌ ሾݔǡ ݕሿሾݔǡ ݖሿ. x ሾݕݔǡ ݖሿ ൌ ሾݔǡ ݖሿሾݕǡ ݖሿ. x ሾ ݔ ǡ ݕሿ ൌ ሾݔǡ ݕሿ .
1 2
ሺషభሻ మ
x ሺݕݔሻ ൌ ݔ ݕ ሾݕǡ ݔሿ
.
Throughout this paper, we consider ܩas a nilpotent group of class two, which means it is not abelian and all the above equations can be applied. Since above equations are successively being used, we apply them without referring. Since ܩൌ ܽۃǡ ܾ ۄis also two generated, then by letting ܿ ൌ ሾܽǡ ܾሿ and applying above equations, we deduce ܿ ൌ ሾܽǡ ܾሿ ൌ ሾܽ ǡ ܾሿ ൌ ܽି ܾ ିଵ ܽ ܾ. Hence,
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International Conference on Mathematical Sciences and Statistics 2013 (ICMSS2013) AIP Conf. Proc. 1557, 41-45 (2013); doi: 10.1063/1.4823871 © 2013 AIP Publishing LLC 978-0-7354-1183-8/$30.00
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ܽ ܾ ൌ ܾܽ ܿ or ܾܽ ൌ ܽ ܾܿ ି .
AUTOMORPHISMS ON ࡳ
(1)
Thus, there are nonnegative integers ݔǡ ݕand ݖ such that ݃ ൌ ܽ ௫ ܾ ௬ ܿ ௭ for every ݃ ܩ א. Next, we give the latest classification for the above ܩൌ ܽۃǡ ܾۄ, when it is of order . In this classification, ߙ and ߚ are considered the least positive ഀ ഁ integers in which ܽ ǡ ܾ ܩ אԢ.
In this section we assume ܩto be one of the groups described in the above classification. Conditions. To distinguish the automorphisms among the maps on ܩ, we use the following conditions (a) and (b) on a map of the form Eq. (3). Let ܿ ൌ ሾܽǡ ܾሿ. If ݔ and ݕ ’s are nonnegative integers, then the map ܽ ܽ ௫భ ܾ ௫మ ܿ ௫య ߠǣ ቄ ܾ ܽ ௬భ ܾ ௬మ ܿ ௬య
The Classification. [4] Let be a prime and
݊ ʹ a positive integer. Every ʹ-generated -group of class exactly two of order corresponds to an ordered ͷ-tuple of integers ሺߙǡ ߚǡ ߛǢ ߩǡ ߪሻǡ such that: i. ߙ ߚ ߛ ͳ, ii. ߙ ߚ ߛ ൌ ݊, iii. Ͳ ߩ ߛ and Ͳ ߪ ߛ, where ሺߙǡ ߚǡ ߛǢ ߩǡ ߪሻ corresponds to the group presented by ം ഀ ܩൌ ܽۃǡ ܾǣሾܽǡ ܾሿ ൌ ሾܽǡ ܾǡ ܽሿ ൌ ሾܽǡ ܾǡ ܾሿ ൌ ͳǡ ܽ ൌ ഐ ഁ ሾܽǡ ܾሿ ǡ ܾ ൌ ሾܽǡ ܾሿ ۄ. Moreover, (1) If ߙ ߚ , then ܩis isomorphic to: a) ሺߙǡ ߚǡ ߛǢ ߩǡ ߛሻ when ߩ ߪ, b) ሺߙǡ ߚǡ ߛǢ ߛǡ ߪሻ when Ͳ ߪ ൏ ߪ ߙ െ ߚ ߩ or ߪ ൏ ߩ ൌ ߛ, c) ሺߙǡ ߚǡ ߛǢ ߩǡ ߪሻ when Ͳ ߪ ൏ ߩ ൏ ݉݅݊ሺߛǡ ߪ ߙ െ ߚሻ. (2) If ߙ ൌ ߚ ߛ, or ߙ ൌ ߚ ൌ ߛ and ʹ, then ܩ is isomorphic to ሺߙǡ ߚǡ ߛǢ ݉݅݊ሺߩǡ ߪሻǡ ߛሻ. (3) If ߙ ൌ ߚ ൌ ߛ and ൌ ʹ, then ܩis isomorphic to: a) ሺߙǡ ߚǡ ߛǢ ݉݅݊ሺߩǡ ߪሻǡ ߛሻ when Ͳ ݉݅݊ሺߩǡ ߪሻ ൏ ߛ െ ͳ, b) ሺߙǡ ߚǡ ߛǢ ߛ െ ͳǡ ߛ െ ͳሻ when ߩ ൌ ߪ ൌ ߛ െ ͳ, c) ሺߙǡ ߚǡ ߛǢ ߛǡ ߛሻ when ݉݅݊ሺߩǡ ߪሻ ߛ െ ͳ and ݉ܽݔሺߩǡ ߪሻ ൌ ߛ. The groups listed in (1)(a) – (3)(c) are pairwise non-isomorphic. In the given classification, the commutators are considered left-normed, for instance ሾܽǡ ܾǡ ܽሿ ൌ ሾሾܽǡ ܾሿǡ ܽሿ. Let ܩൌ ܽۃǡ ܾ ۄbe one of the groups listed in the classification. Then, ȁܽȁ ൌ ఈାఊିఘ and ȁܾȁ ൌ ఉାఊିఙ .
(2)
(3)
can be extended to an automorphism on ܩif and only if both of the following conditions hold: ݔଵ ݔଶ a) ݀ሺߠሻ ൌ ቚ ݕ ݕቚ ൌ ݔଵ ݕଶ െ ݔଶ ݕଵ Ͳ ء, ଵ ଶ (mod ). ഀ
ഐ
ഁ
b) ൫ߠሺܽሻ൯ ൌ ሺሾߠሺܽሻǡ ߠሺܾሻሿሻ ǡ ൫ߠሺܾሻ൯ ൌ ം ሺሾߠሺܽሻǡ ߠሺܾሻሿሻ and ሾߠሺܽሻǡ ߠሺܾሻሿ ൌ ͳ. If ߠ extends to an automorphism, then (b) clearly holds. Also, ߠ induces an automorphism ߠҧ on ܩΤ߶ሺܩሻ by ߠҧ൫݃߶ሺܩሻ൯ ൌ ߠሺ݃ሻ߶ሺܩሻ. We know that ܩΤ߶ሺܩሻ is a vector space of dimension ʹ over Ժ and ܿ ܩ אԢ ك ߶ሺܩሻ [5]. Thus ߠҧ satisfies (a) and so does ߠ. Conversely, (b) shows that ߠ can be extended to a unique endomorphism of ܩ. Moreover, by (a) we find that ܩΤ߶ሺܩሻ ൌ ߠۃሺܽሻ߶ሺܩሻǡ ߠሺܾሻ߶ሺܩሻۄ, which implies ܩൌ ߠۃሺܽሻǡ ߠሺܾሻ[ ۄ3].
Inner automorphisms. Let ߠ maps ሼܽǡ ܾሽ to ܩas follows:
ܽ ܽܿ ௫ ߠǣ ൜ ܾ ܾܿ ௬ ǡ
(4)
where Ͳ ݔǡ ݕ൏ ఊ . Then ߠ is extendable to an automorphism on ܩ. Indeed, Condition (b) holds since ሾܽܿ ௫ ǡ ܾܿ ௬ ሿ ൌ ܿ and ߙǡ ߚ ߛ. Also, ݀ሺߠሻ ൌ ͳ shows that ߠ satisfies (a). In [3] we have shown that the inner group of ܩ, ݊݊ܫሺܩሻ, consists of all automorphisms of the form Eq. (4). Hence, the following map extends to a unique automorphism on ܩ: ܽ ܽ ௫భ ܾ ௫మ ߠǣ ቄ (5) ܾ ܽ ௬భ ܾ ௬మ if and only if it satisfies Conditions (a) and (b).
MAIN RESULTS Let ܩbe a group as described in the classification, which is presented by ሺߙǡ ߚǡ ߛǢ ߩǡ ߪሻ. If ߠ is an automorphism on ܩof the form Eq. (5) that induces
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identity transformation on ܩΤ߶ሺܩሻ, then ߠҧ ൌ ݅Ǥ In other words, we should have ߠҧ൫ܽ߶ሺܩሻ൯ ൌ ߠሺܽሻ߶ሺܩሻ ൌ ܽ߶ሺܩሻ and ߠҧ൫ܾ߶ሺܩሻ൯ ൌ ߠሺܾሻ߶ሺܩሻ ൌ ܾ߶ሺܩሻǤ Henceܽ ିଵ ߠሺܽሻǡ ܾ ିଵ ߠሺܾሻ ߶ אሺܩሻǤ However ߶ሺܩሻ ൌ൏ ܽ ǡ ܾ ǡ ܿ , thus ߠሺܽሻ ൌ ܽܽ௫భ ܾ ௫మ ܿ ௫య and ߠሺܾሻ ൌ ܾܽ௬భ ܾ ௬మ ܿ ௬య for some ݔ ݕ ƍݏ. By applying Eq. (1), we find ߠሺܽሻ ൌ ܽ ሺ௫భାଵሻ ܾ ௫మ ܿ ௫య ൜ ߠሺܾሻ ൌ ܽ ௬భ ܾ ሺ௬మାଵሻ ܿ ௬య Ǥ Substituting the above relations in Condition (b), leads to the following equation: ഀశభ
ഐశభ
ܾ ௫మ ൌ ܿ ሺௗା௬మሻ ቊ ഁశభ௬ శభ ሺௗା௫ ሻ భ ൌ ܿ భ ǡ ܽ
(6)
where ൌ ݀ሺߠሻ , Ͳ ݔଵ ǡ ݕଵ ൏ ఈ and Ͳ ݔଶ ǡ ݕଶ ൏ ఉ . Therefore, if ߠ is an automorphism on ܩthat induces the identity transformation on ܩΤ߶ሺܩሻ, then it should satisfy Eq. (6).
Finding The Limitations. Let ܩbe a two
generated group of nilpotency class two and order that is presented by ሺߙǡ ߚǡ ߛǢ ߩǡ ߪሻ according to the classification. Let ߠ be an automorphism on ܩgiven by Eq. (5). Put ݉ ൌ ߛ െ ߩ and ݇ ൌ ߛ െ ߪ. We consider to find limitations on ݔଵ ǡ ݔଶ ǡ ݕଵ and ݕଶ which make ߠ to satisfy Eq. (6). As a matter of fact, we will have ߠҧ ൌ ݅ under these limitation. To accomplish so, we study three cases ߪ ߛ െ ͳ, ߪ ൌ ߛ െ ʹ and ߪ ൏ ߛ െ ʹ. For each case, we first give the limitations on ݔ and ݕ ’s in several steps. Next, the proof for each step will be given in order. Case A. Suppose ߪ ߛ െ ͳ.
A.1) Let ߩ ൌ ߛ. A.1.1. If ߚ ߙ ߚ ͳ, then we have no condition. A.1.2. If ߙ ߚ ͳ, then ሺఈିఉିଵሻ ȁݕଵ . A.2) Let ߩ ൌ ߛ െ ͳ. A.2.1. If ߙ ൌ ߚ, then we have no condition. A.2.2. If ߙ ߚ, then ሺఈିఉሻ ȁݕଵ . A.3) Let ߩ ߛ െ ʹ. A.3.1. If ߙ ൌ ߚ, then ሺିଵሻ ȁݕଵ ; ȁݕଶ and if ݉ ʹ then ଶ ݕ ץଶ . A.3.2. If ߙ ߚ, then ሾሺఈିఉሻାሺିଵሻሿ ȁݕଵ ; ȁݕଶ and if ݉ ʹ then ଶ ݕ ץଶ . Proof of Case (A). Part 1. Let ߪ ൌ ߛ.
A.1) For all ߙ ߚ we have ܩሺߙǡ ߚǡ ߛǢ ߛǡ ߛሻ. Thus ഀ ഁ ം ܽ ൌ ܾ ൌ ܿ ൌ ͳ. Hence Eq. (6) implies ഀశభ ഁశభ ܾ ௫మ ൌ ͳ. Also, we find ܽ ௬భ ൌ ͳ which implies ఈ ఉାଵ ሺఈିఉିଵሻ ȁݕଵ if ߙ ߚ ͳ. ȁݕ ଵ or A.2) We know that ܩሺߙǡ ߚǡ ߛǢ ߛ െ ͳǡ ߛሻ if ߙ ߚ, ߙ ൌ ߚ ߛ or ߙ ൌ ߚ ൌ ߛ and ʹ. In all these cases ഀశభ ഁ ം we have ܽ ൌ ܾ ൌ ܿ ൌ ͳ. So, regarding Eq. (6) we find ఈାଵ ȁఉାଵ ݕଵ or ఈିఉ ȁݕଵ when ߙ ߚ. If ߙ ൌ ߚ ൌ ߛ and ൌ ʹ, then ܩሺߛǡ ߛǡ ߛǢ ߛǡ ߛሻ. Thus, there is no condition. A.3) Let ߩ ߛ െ ʹ. Then for all ߙ ߚ we have ܩሺߙǡ ߚǡ ߛǢ ߩǡ ߛሻ. By Eq. (2) we have ȁܽȁ ൌ ሺఈାఊିఘሻ ൌ ఈା , thus ఈା ȁఉାଵ ݕଵ or ሾሺఈିఉሻାሺିଵሻሿ ఊ ሺఘାଵሻ ȁݕଵ . Also, ȁ ሺ ݀ ݕଶ ሻ implies ȁݕଶ . In case ߩ ൏ ߛ െ ʹ we have ݉ ʹ, and since ݀ ץ , we may write ଶ ݕ ץଶ . Part 2. Let ߪ ൌ ߛ െ ͳ. A.1.1. If ߙ ൌ ߚ ߛ or ߙ ൌ ߚ ൌ ߛ and ʹ, then ഀశభ ܩሺߙǡ ߙǡ ߛǢ ߛ െ ͳǡ ߛሻ. This shows that ܽ ൌ ഀ ം ܾ ൌ ܿ ൌ ͳ. By Eq. (6) we find ͳ ൌ ͳ. Hence, we find no condition. Also, if ߙ ൌ ߚ ൌ ߛ and ൌ ʹ, then ܩ ሺߛǡ ߛǡ ߛǢ ߛǡ ߛሻ. Again we have no condition. A.1.2. If ߙ ߚ then ܩሺߙǡ ߚǡ ߛǢ ߛǡ ߛ െ ͳሻ. Thus ഀ ഁశభ ം ܽ ൌ ܾ ൌ ܿ ൌ ͳ. Hence Eq. (6) implies ഀశభ ഁశభ ܾ ௫మ ൌ ͳ . Also, ܽ ௬భ ൌ ͳ again leads to ሺఈିఉିଵሻ ȁݕଵ if ߙ ߚ ͳ. A.2) ܩሺߙǡ ߚǡ ߛǢ ߛ െ ͳǡ ߛሻ if ߙ ߚ, ߙ ൌ ߚ ߛ or ߙ ൌ ߚ ൌ ߛ and ʹ. Anyway, it is similar to the proof of A.2 in Part 1 of Case (A). If ߙ ൌ ߚ ൌ ߛ and ൌ ʹ, then ܩሺߛǡ ߛǡ ߛǢ ߛ െ ംశభ ംశభ ം ͳǡ ߛ െ ͳሻ. Thus ܽ ൌ ܾ ൌ ܿ ൌ ͳ. Thus, Eq. (6) obviously holds. A.3) It is completely similar to A.3 in Part 1 of Case (A). Case B. Suppose ߪ ൌ ߛ െ ʹ. B.1) Let ߩ ൌ ߛ. B.1.1. If ߙ ൌ ߚ, then ȁݕଵ and ȁݕଶ . B.1.2. If ߙ ൌ ߚ ͳ, then ȁݔଵ . B.1.3. If ߙ ൌ ߚ ʹ, then we have no condition. B.1.4. If ߙ ߚ ʹ, then ሺఈିఉିଶሻ ȁݕଵ . B.2) Let ߩ ൌ ߛ െ ͳ. B.2.1. If ߙ ൌ ߚ, then ȁݕଵ and ȁݕଶ . B.2.2. If ߙ ൌ ߚ ͳ, then ȁݔଵ .
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B.2.3. If ߙ ߚ ͳ, then ሺఈିఉିଵሻ ȁݕଵ . B.3) Let ߩ ߛ െ ʹ. B.3.1. If ߙ ൌ ߚ, then ିଵ ȁݕଵ , ȁݕଶ and if ݉ ʹ then ଶ ݕ ץଶ . B.3.2. If ߙ ߚ, then ሾሺఈିఉሻାሺିଵሻሿ ȁݕଵ ; ȁݕଶ and if ݉ ʹ then ଶ ݕ ץଶ . Proof of Case (B). B.1.1. If ߙ ൌ ߚ, then ܩሺߙǡ ߙǡ ߛǢ ߛ െ ʹǡ ߛሻ. Hence ഀశమ ഀ ം ܽ ൌ ܾ ൌ ܿ ൌ ͳ. By applying Eq. (6) we find ഀశభ ംషభ ܽ ௬భ ൌ ͳ and ܿ ௬మ ൌ ͳ. So, ఈାଶ ȁఈାଵ ݕଵ or ȁݕଵ . Additionally, ఊ ȁఊିଵ ݕଶ implies ȁݕଶ . If we consider ߙ ߚ, then we will find ܩ ሺߙǡ ߚǡ ߛǢ ߛǡ ߛ െ ʹሻ. Hence, in all steps B.1.2, B.1.3 and B.1.4 we consider ܩto be presented as given. We also ഀ ഁశమ ം have ܽ ൌ ܾ ൌ ܿ ൌ ͳ. By using Eq. (6), we ഀశభ find ܾ ௫మ ൌ ͳ which is trivial in all these steps since ߙ ͳ ߚ ʹ. B.1.2. If ߙ ൌ ߚ ͳ, then we have ܽ ംషభ ܿ ௫భ ൌ ͳ which implies ȁݔଵ . B.1.3. If ߙ ൌ ߚ ʹ, then we find ܽ ͳ. So, we find no condition.
ഁశభ ௬భ
ൌ
ൌ
ഁశమ ௬ భ
ൌ ܿ
ം௫ భ
ഁశభ ௬ భ
ൌ ܿ
ംషభ ௫ భ
B.1.4. If ߙ ߚ ʹ, then ܽ Therefore, ఈ ȁఉାଶ ݕଵ or ሺఈିఉିଶሻ ȁݕଵ .
.
B.2.1. If ߙ ൌ ߚ, then ܩሺߙǡ ߙǡ ߛǢ ߛ െ ʹǡ ߛሻ. Then it is similar to the proof of B.1.1. B.2.2. If ߙ ൌ ߚ ͳ, then ܩሺߚ ͳǡ ߚǡ ߛǢ ߛǡ ߛ െ ʹሻ. ഁశభ ഁశమ ം Thus ܽ ൌ ܾ ൌ ܿ ൌ ͳ. By using Eq. (6) we ഀశభ ഁశమ find ܾ ௫మ ൌ ܾ ௫మ ൌ ͳ which leads to no ഁశభ ംషభ condition. Additionally, we have ܽ ௬భ ൌ ܿ ௫భ ൌ ͳ, which implies ఊ ȁఊିଵ ݔଵ or ȁݔଵ . B.2.3. If ߙ ߚ ͳ, then ܩሺߙǡ ߚǡ ߛǢ ߛ െ ͳǡ ߛ െ ʹሻ. ഀశభ ഁశమ ം ഁశభ Thus ܽ ൌ ܾ ൌ ܿ ൌ ͳ. Hence ܽ ௬భ ൌ ംషభ ܿ ௫భ leads to ఈାଵ ȁఉାଶ ݕଵ or ሺఈିఉିଵሻ ȁݕଵ . B.3. For all ߙ and ߚ we have ܩሺߙǡ ߚǡ ߛǢ ߩǡ ߛሻ. Thus, ഀశ ഁ ം ܽ ൌ ܾ ൌ ܿ ൌ ͳ. Using Eq. (6) leads to ఈା ȁఉାଵ ݕଵ and ିଵ ȁ ݀ ݕଶ . Thus, we obtain ሾሺఈିఉሻାሺିଵሻሿ ȁݕଵ and ȁݕଶ . If ݉ ʹ then ଶ ݕ ץଶ . Case C. Suppose ߪ ൏ ߛ െ ʹ.
C.1.1. If ߙ ൌ ߚ, then ିଵ ȁݕଵ ; ȁݕଶ and ଶ ݕ ץଶ . C.1.2. If ߚ ൏ ߙ ൏ ߚ ݇ െ ͳ, then ሾሺିଵሻିሺఈିఉሻሿ ȁݔଶ ; ȁݔଵ and ଶ ݔ ץଵ . C.1.3. If ߙ ൌ ߚ ݇ െ ͳ, then ȁݔଵ . C.1.4. If ߙ ൌ ߚ ݇, there is no condition. C.1.5. If ߙ ߚ ݇, then ሺఈିఉିሻ ȁݕଵ . C.2) Let ߩ ൌ ߛ െ ͳ. C.2.1. If ߙ ൌ ߚ, then ିଵ ȁݕଵ ; ȁݕଶ and ଶ ݕ ץଶ . C.2.2. If ߚ ൏ ߙ ൏ ߚ ݇ െ ͳ, then ሾሺିଵሻିሺఈିఉሻሿ ȁݔଶ ; ȁݔଵ and ଶ ݔ ץଵ . C.2.3. If ߙ ൌ ߚ ݇ െ ͳ, then ȁݔଵ . C.2.4. If ߙ ߚ ݇ െ ͳ, then ሾሺఈିఉሻାሺିଵሻሿ ȁݕଵ . C.3) Let ߩ ൌ ߛ െ ʹ. C.3.1. If ߙ ൌ ߚ, then ିଵ ȁݕଵ ; ȁݕଶ and ଶ ݕ ץଶ . C.3.2. If ߚ ൏ ߙ ൏ ߚ ݇ െ ͳ, then ሾሺିଵሻିሺఈିఉሻሿ ȁݔଶ ; ȁݔଵ and ଶ ݔ ץଵ . C.3.3. If ߙ ߚ ݇ െ ͳ, then ሾሺఈିఉሻାሺିଶሻሿ ȁݕଵ and ȁݕଶ . C.4) Let ߩ ൏ ߛ െ ʹ. C.4.1. If ߙ ൌ ߚ, then ିଵ ȁݕଵ ; ȁݕଶ and ଶ ݕ ץଶ . C.4.2. Let ߙ ߚ and ߩ ߪ. Then ሾሺఈିఉሻାሺିଵሻሿ ȁݕଵ ; ȁݕଶ and ଶ ݕ ץଶ . C.4.3. Let ߙ ߚ and Ͳ ߪ ൏ ߪ ߙ െ ߚ ߩ. Then ሾሺିଵሻିሺఈିఉሻሿ ȁݔଶ ; ȁݔଵ and ଶ ݔ ץଵ . C.4.4. Let ߙ ߚ and Ͳ ߪ ൏ ߩ ൏ ሺߛǡ ߪ ߙ െ ߚሻ. Then ሾሺఈିఉሻିሺିሻሿ ȁݕଵ and ȁݕଶ . Also, if ሺߙ െ ߚሻ െ ሺ݇ െ ݉ሻ ͳ, then ଶ ݕ ץଶ . Proof of Case (C). C.1.1., C.2.1., C.3.1., C.4.1. If ߙ ൌ ߚ, then ܩ ഀశ ሺߙǡ ߙǡ ߛǢ ߩǡ ߛሻ, where ߩ ൌ ሺ ߩǡ ߪሻ. Hence ܽ ൌ ഀ ം ܾ ൌ ܿ ൌ ͳ. By applying Eq. (6) we obtain ഀశభ ഐశభ ܽ ௬భ ൌ ͳ and ܿ ሺௗା௬మ ሻ ൌ ͳ. Therefore, ഀశ ഀశభ ȁܽ ௬భ or ିଵ ȁݕଵ . Additionally, ܽ ఊ ఘାଵ ȁ ሺ ݀ ݕଶ ሻ implies ȁݕଶ and ଶ ݕ ץଶ since ߩ ൏ ߛ െ ʹ. Let ߙ ߚ. If ߩ ൌ ߛ or Ͳ ߪ ൏ ߪ ߙ െ ߚ ߩ or equivalently ߚ ൏ ߙ ߚ ݇ െ ͳ, then we have ܩሺߙǡ ߚǡ ߛǢ ߛǡ ߪሻ. Hence, ݇ ൌ ߛ െ ߪ ʹ and ഀ ഁశೖ ം ܽ ൌ ܾ ൌ ܿ ൌ ͳ. C.1.2., C.2.2., C.3.2., C.4.3. Let ߚ ൏ ߙ ൏ ߚ ݇ െ ͳ. ഀశభ ഁశభ Using Eq. (6) leads to ܾ ௫మ ൌ ͳ and ܽ ௬భ ൌ శభ Thus, ఉା ȁఈାଵ ݔଶ and ܿ ሺௗା௫భሻ . శሺഀషഁሻ ሺௗା௫భ ሻ ൌ ͳ. Finally, ሾሺିଵሻିሺఈିఉሻሿ ȁݔଶ and ܿ ȁݔଵ while as ଶ ݔ ץଵ , since ݇ െ ሺߙ െ ߚሻ ʹ.
C.1) Let ߩ ൌ ߛ.
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C.1.3., C.2.3. Let ߙ ൌ ߚ ݇ െ ͳ. Regarding previous step, we only obtain ȁݔଵ . C.1.4., C.1.5. If ߙ ߚ ݇ െ ͳ, then trivially ഁశభ శభ ఉା ȁఈାଵ ݔଶ . Thus ܽ ௬భ ൌ ܿ ሺௗା௫భ ሻ is the ഁశೖ only remainder condition, which leads to ܽ ௬భ ൌ ͳ or ఈ ȁఉା ݕଵ . Hence we have no condition if ߙ ൌ ߚ ݇ and ሺఈିఉିሻ ȁݕଵ in case ߙ െ ߚ ݇. C.2.4. Let ߙ ߚ ݇ െ ͳ. In this state ܩ ഀశభ ഁశೖ ം ሺߙǡ ߚǡ ߛǢ ߛ െ ͳǡ ߪሻ. Hence, ܽ ൌ ܾ ൌ ܿ ൌ ͳ. ഁశభ శభ Therefore, ܽ ௬భ ൌ ܿ ሺௗା௫భ ሻ implies ሾሺఈିఉሻିሺିଵሻሿ ఈାଵ ఉା ȁ ݕଵ or ȁݕଵ .
REFERENCES 1. Y. Barakat and N. H. Sarmin, “Automorphisms of 2generated p-groups with cyclic commutator subgroup”, Proceedings of International Seminar on the Application of Science & Mathematics, No.83. MTH018 (2011). 2. L. C. Kappe, M. P. Visscher, and N. H. Sarmin, Twogenerator two-groups of class two and their nonabelian tensor squares. Glasgow Math. J.,41, 417-430, 1999. 3. D. Gorentein, Finite groups, (Second ed.) Chelsea Pub Co, 2007. 4. A. Ahmad, A. Magidin, and R. F. Morse. Two generator p-groups of nilpotency class 2 and their conjugacy classes. Publ Math-Debrecen, 81: 1-2(10) (2012).
C.3.3. Let ߙ ߚ ݇ െ ͳ. Then ܩሺߙǡ ߚǡ ߛǢ ߛ െ ʹǡ ߪሻ. The proof details are similar to C.2.4. when ഁశೖశభ ߙ ߚ ݇ െ ͳ. If ߙ ൌ ߚ ݇ െ ͳ, then ܽ ൌ ഁశೖ ം ܾ ൌ ܿ ൌ ͳ. Thus ఉାାଵ ȁఉା ݕଵ or ȁݕଵ . C.4.2. Let ߙ ߚ and ߩ ߪ. Then ܩሺߙǡ ߚǡ ߛǢ ߩǡ ߛሻ. ഁశభ Applying Eq. (6) leads to ܽ ௬భ ൌ ͳ and ഐశభ ഀశ ഁశభ ܿ ሺௗା௬మሻ ൌ ͳ. Thus, ܽ ȁܽ ௬భ and ఊ ȁఘାଵ ሺ ݀ ݕଶ ሻ. Therefore, ሾሺఈିఉሻାሺିଵሻሿ ȁݕଵ and ȁݕଶ but ଶ ݕ ץଶ since ݉ ʹ. C.4.4. If ߙ ߚ and Ͳ ߪ ൏ ߩ ൏ ሺߛǡ ߪ ߙ െ ߚሻ, then ܩሺߙǡ ߚǡ ߛǢ ߩǡ ߪሻ. In this step we have ݇ െ ݉ ൌ ߩ െ ߪ ൏ ߙ െ ߚ or ݇ െ ሺߙ െ ߚሻ ൏ ݉. Also, ഀశ ഁశೖ ം ܽ ൌ ܾ ൌ ܿ ൌ ͳ. Based on Eq. (6), we find ሾሺೖషభሻషሺഀషഁሻሿ ഁశೖ ഀశభ ൌ ሺܾ ௫మ ሻ ൌ ͳ ൌ ܾ ഐశభ ሺௗା௬మ ሻ ሾሺೖషభሻషሺഀషഁሻሿ ሺܿ ሻ which implies ఊ ȁሾሺାఘሻିሺఈିఉሻሿ ሺ ݀ ݕଶ ሻ. Since ߛ െ ݇ െ ߩ ൌ ݉ െ ݇, we obtain ሾሺఈିఉሻିሺିሻሿ ȁሺ ݀ ݕଶ ሻ which means ȁݕଶ . If ሺߙ െ ߚሻ െ ሺ݇ െ ݉ሻ ͳ, then we may add ଶ ݕ ץଶ . ഁశభ ೖషభ శభ ೖషభ ൌ ሺܿ ሺௗା௫భሻ ሻ ൌ Moreover, ሺܽ ௬భ ሻ ఈା ఉା ȁ ݕଵ . The latest expression equals ͳ implies ഀశభ షభ ൌ to ሾሺఈିఉሻିሺିሻሿ ȁݕଵ . Note that ሺܾ ௫మ ሻ ഐశభ షభ ሺܿ ሺௗା௬మ ሻ ሻ ൌ ͳ leads to ఉା ȁఈା ݔଶ which is trivial in this step. Similarly, we may find ሾሺഀషഁሻశሺషభሻሿ ഀశభ శభ ͳ ൌ ܽ ௬భ ൌ ሺܿ ሺௗା௫భ ሻ ሻ , which is an obvious equivalency.
ACKNOWLEDGMENTS The authors would like to express their sincerest appreciation to Prof. V. D. Mazurov for his valuable hints and suggestions. Also, the second author would like to thank Universiti Teknologi Malaysia for the financial support via allocating International Doctoral Fellowship.
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