Specific automorphisms on a 2-generated p-group of

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 ԺΤ‫݌‬Ժ. 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 ԺΤ‫݌‬Ժ, 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

೘ሺ೘షభሻ 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


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 ൌ ܽ ሺ௣௫భାଵሻ ܾ ௣௫మ ܿ ௫య ൜ ߠሺܾሻ ൌ ܽ ௣௬భ ܾ ሺ௣௬మାଵሻ ܿ ௬య Ǥ Substituting the above relations in Condition (b), leads to the following equation: ഀశభ

ഐశభ

ܾ ௣ ௫మ ൌ ܿ ௣ ሺ௣ௗା௬మሻ ቊ ௣ഁశభ௬ ഑శభ ሺ௣ௗା௫ 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 ‫ ݌‬ሺ௠ିଵሻ ȁ‫ݕ‬ଵ ; ‫݌‬ȁ‫ݕ‬ଶ and if ݉ ൐ ʹ then ‫݌‬ଶ ‫ݕ ץ‬ଶ . A.3.2. If ߙ ൐ ߚ, then ‫ ݌‬ሾሺఈିఉሻାሺ௠ିଵሻሿ 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 ȁ‫݌‬ఉାଵ ‫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௠ ȁ‫݌‬ఉାଵ ‫ݕ‬ଵ or ሾሺఈିఉሻାሺ௠ିଵሻሿ 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 .


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 ‫݌‬ሾሺఈିఉሻାሺ௠ିଵሻሿ 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 ȁ‫݌‬ఈାଵ ‫ݕ‬ଵ or ‫݌‬ȁ‫ݕ‬ଵ . Additionally, ‫݌‬ఊ ȁ‫݌‬ఊିଵ ‫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 ȁ‫݌‬ఉାଶ ‫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 ȁ‫݌‬ఊିଵ ‫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 ȁ‫݌‬ఉାଶ ‫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௠ ȁ‫݌‬ఉାଵ ‫ݕ‬ଵ and ‫݌‬௠ିଵ ȁ‫ ݀݌‬൅ ‫ݕ‬ଶ . Thus, we obtain ‫݌‬ሾሺఈିఉሻାሺ௠ିଵሻሿ 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 ‫݌‬ሾሺ௞ିଵሻିሺఈିఉሻሿ 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 ‫ ݌‬ሺఈିఉି௞ሻ 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 ‫݌‬ሾሺ௞ିଵሻିሺఈିఉሻሿ ȁ‫ݔ‬ଶ ; ‫݌‬ȁ‫ݔ‬ଵ and ‫݌‬ଶ ‫ݔ ץ‬ଵ . C.2.3. If ߙ ൌ ߚ ൅ ݇ െ ͳ, then ‫݌‬ȁ‫ݔ‬ଵ . C.2.4. If ߙ ൐ ߚ ൅ ݇ െ ͳ, then ‫݌‬ሾሺఈିఉሻାሺ௞ିଵሻሿ 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 ‫݌‬ሾሺ௞ିଵሻିሺఈିఉሻሿ ȁ‫ݔ‬ଶ ; ‫݌‬ȁ‫ݔ‬ଵ and ‫݌‬ଶ ‫ݔ ץ‬ଵ . C.3.3. If ߙ ൒ ߚ ൅ ݇ െ ͳ, then ‫݌‬ሾሺఈିఉሻାሺ௞ିଶሻሿ 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 ‫݌‬ሾሺఈିఉሻାሺ௠ିଵሻሿ ȁ‫ݕ‬ଵ ; ‫݌‬ȁ‫ݕ‬ଶ and ‫݌‬ଶ ‫ݕ ץ‬ଶ . C.4.3. Let ߙ ൐ ߚ and Ͳ ൑ ߪ ൏ ߪ ൅ ߙ െ ߚ ൑ ߩ. Then ‫݌‬ሾሺ௞ିଵሻିሺఈିఉሻሿ ȁ‫ݔ‬ଶ ; ‫݌‬ȁ‫ݔ‬ଵ and ‫݌‬ଶ ‫ݔ ץ‬ଵ . C.4.4. Let ߙ ൐ ߚ and Ͳ ൑ ߪ ൏ ߩ ൏ ሺߛǡ ߪ ൅ ߙ െ ߚሻ. Then ‫ ݌‬ሾሺఈିఉሻିሺ௞ି௠ሻሿ 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 ܿ ௣ ሺ௣ௗା௬మ 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௞ ȁ‫݌‬ఈାଵ ‫ݔ‬ଶ and ܿ ௣ ሺ௣ௗା௫భሻ . ௣഑శሺഀషഁሻ ሺ௣ௗା௫భ ሻ ൌ ͳ. Finally, ‫ ݌‬ሾሺ௞ିଵሻିሺఈିఉሻሿ ȁ‫ݔ‬ଶ and ܿ ‫݌‬ȁ‫ݔ‬ଵ while as ‫݌‬ଶ ‫ݔ ץ‬ଵ , since ݇ െ ሺߙ െ ߚሻ ൒ ʹ.

C.1) Let ߩ ൌ ߛ.


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௞ ȁ‫݌‬ఈାଵ ‫ݔ‬ଶ . Thus ܽ௣ ௬భ ൌ ܿ ௣ ሺ௣ௗା௫భ ሻ is the ഁశೖ only remainder condition, which leads to ܽ ௣ ௬భ ൌ ͳ or ‫݌‬ఈ ȁ‫݌‬ఉା௞ ‫ݕ‬ଵ . Hence we have no condition if ߙ ൌ ߚ ൅ ݇ and ‫݌‬ሺఈିఉି௞ሻ 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 ൌ ܿ ௣ ሺ௣ௗା௫భ ሻ implies ሾሺఈିఉሻିሺ௞ିଵሻሿ 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 ‫݌‬ఉା௞ାଵ ȁ‫݌‬ఉା௞ ‫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 ܿ ௣ ሺ௣ௗା௬మሻ ൌ ͳ. Thus, ܽ௣ ȁܽ ௣ ௬భ and ‫݌‬ఊ ȁ‫݌‬ఘାଵ ሺ‫ ݀݌‬൅ ‫ݕ‬ଶ ሻ. Therefore, ‫݌‬ሾሺఈିఉሻାሺ௠ିଵሻሿ 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 ሺ௣ௗା௬మ ሻ ௣ሾሺೖషభሻషሺഀషഁሻሿ ሺܿ ሻ which implies ‫݌‬ఊ ȁ‫݌‬ሾሺ௞ାఘሻିሺఈିఉሻሿ ሺ‫ ݀݌‬൅ ‫ݕ‬ଶ ሻ. Since ߛ െ ݇ െ ߩ ൌ ݉ െ ݇, we obtain ‫ ݌‬ሾሺఈିఉሻିሺ௞ି௠ሻሿ ȁሺ‫ ݀݌‬൅ ‫ݕ‬ଶ ሻ which means ‫݌‬ȁ‫ݕ‬ଶ . If ሺߙ െ ߚሻ െ ሺ݇ െ ݉ሻ ൐ ͳ, then we may add ‫݌‬ଶ ‫ݕ ץ‬ଶ . ഁశభ ೖషభ ഑శభ ೖషభ ൌ ሺܿ ௣ ሺ௣ௗା௫భሻ ሻ௣ ൌ Moreover, ሺܽ ௣ ௬భ ሻ௣ ఈା௠ ఉା௞ ȁ‫݌‬ ‫ݕ‬ଵ . The latest expression equals ͳ implies ‫݌‬ ഀశభ ೘షభ ൌ to ‫݌‬ሾሺఈିఉሻିሺ௞ି௠ሻሿ ȁ‫ݕ‬ଵ . Note that ሺܾ ௣ ௫మ ሻ௣ ഐశభ ೘షభ ሺܿ ௣ ሺ௣ௗା௬మ ሻ ሻ௣ ൌ ͳ leads to ‫݌‬ఉା௞ ȁ‫݌‬ఈା௠ ‫ݔ‬ଶ which is trivial in this step. Similarly, we may find ሾሺഀషഁሻశሺ೘షభሻሿ ഀశభ ഑శభ ͳ ൌ ܽ௣ ௬భ ൌ ሺܿ ௣ ሺ௣ௗା௫భ 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.
